AACIMP 2010 Summer School lecture by Vsevolod Vladimirov. "Applied Mathematics" stream. "Selected Models of Transport Processes. Methods of Solving and Properties of Solutions" course. Part 2.
More info at http://summerschool.ssa.org.ua
Nonlinear transport phenomena: models, method of solving and unusual features...SSA KPI
AACIMP 2010 Summer School lecture by Vsevolod Vladimirov. "Applied Mathematics" stream. "Selected Models of Transport Processes. Methods of Solving and Properties of Solutions" course. Part 3.
More info at http://summerschool.ssa.org.ua
We consider a general approach to describing the interaction in multigravity models in a D-dimensional space–time. We present various possibilities for generalizing the invariant volume. We derive the most general form of the interaction potential, which becomes a Pauli–Fierz-type model in the bigravity case. Analyzing this model in detail in the (3+1)-expansion formalism and also requiring the absence of ghosts leads to this bigravity model being completely equivalent to the Pauli–Fierz model. We thus in a concrete example show that introducing an interaction between metrics is equivalent to introducing the graviton mass.
This document summarizes a presentation on numerically solving spatiotemporal models from ecology by implementing an implicit finite difference method in C++. It discusses classical population dynamics models, extending these to include continuous spatial position by using reaction-diffusion systems. It describes discretizing the domain and deriving finite-difference schemes, then solving the equations using GMRES with an ILU preconditioner. Questions addressed include convergence rates and modeling plankton dynamics.
This presentation illustrates the principles of thermodynamics in the freezing soil according to the capillary schematization and the freezing=drying assumption
The document summarizes a lecture on the Ornstein-Uhlenbeck process. It describes how Ornstein and Uhlenbeck introduced this process in the 1930s as a more accurate model of Brownian motion than previous models. The process is defined as the unique solution to the stochastic differential equation dY(t) = -KY(t)dt + dX(t), where X(t) is a driving Lévy process and K is a matrix. Key properties are that Y(t) has càdlàg paths and is a Markov process. The process generalizes previous models where X(t) was Brownian motion.
What happens when the Kolmogorov-Zakharov spectrum is nonlocal?Colm Connaughton
This document summarizes research on the behavior of the Kolmogorov-Zakharov (KZ) spectrum when it is nonlocal. It examines a model of cluster-cluster aggregation described by the Smoluchowski equation, which can be viewed as a model of 3-wave turbulence without backscatter. The research finds that when the exponents in the interaction term satisfy certain conditions, the KZ spectrum is nonlocal. In this case, the stationary state has a novel functional form and can become unstable, leading to oscillatory behavior in the cascade dynamics at long times. Open questions remain about whether physical systems exhibit this behavior and how the results are affected by including backscatter terms.
kinks and cusps in the transition dynamics of a bloch statejiang-min zhang
We discuss the transition dynamics of a Bloch state in a 1D tight binding chain, under two scenarios, namely, weak periodical driving [1] and sudden quench [2]. In the former case, the survival probability of the initial Bloch state shows kinks periodically; it is a piece-wise linear function of time. In the latter, the survival probability (Loschmidt echo in this case) shows cusps periodically; it is a piece-wise quadratic function of time. Kinks in the former case are a perturbative effect, while cusps in the latter are a non-perturbative effect. The kinks and cusps are reminiscent of the so-called dynamical phase transtion termed by Heyl et al. [3].
[1] J. M. Zhang and M. Haque, Nonsmooth and level-resolved dynamics illustrated with the tight binding model, arXiv:1404.4280.
[2] J. M. Zhang and H. T. Yang, Cusps in the quenched dynamics of a bloch state, arXiv:1601.03569.
[3] M. Heyl, A. Polkovnikov, and S. Kehrein, Dynamical quantum phase transitions in the transverse-field Ising model, Phys. Rev. Lett. 110, 135704 (2013).
Nonlinear transport phenomena: models, method of solving and unusual features...SSA KPI
AACIMP 2010 Summer School lecture by Vsevolod Vladimirov. "Applied Mathematics" stream. "Selected Models of Transport Processes. Methods of Solving and Properties of Solutions" course. Part 3.
More info at http://summerschool.ssa.org.ua
We consider a general approach to describing the interaction in multigravity models in a D-dimensional space–time. We present various possibilities for generalizing the invariant volume. We derive the most general form of the interaction potential, which becomes a Pauli–Fierz-type model in the bigravity case. Analyzing this model in detail in the (3+1)-expansion formalism and also requiring the absence of ghosts leads to this bigravity model being completely equivalent to the Pauli–Fierz model. We thus in a concrete example show that introducing an interaction between metrics is equivalent to introducing the graviton mass.
This document summarizes a presentation on numerically solving spatiotemporal models from ecology by implementing an implicit finite difference method in C++. It discusses classical population dynamics models, extending these to include continuous spatial position by using reaction-diffusion systems. It describes discretizing the domain and deriving finite-difference schemes, then solving the equations using GMRES with an ILU preconditioner. Questions addressed include convergence rates and modeling plankton dynamics.
This presentation illustrates the principles of thermodynamics in the freezing soil according to the capillary schematization and the freezing=drying assumption
The document summarizes a lecture on the Ornstein-Uhlenbeck process. It describes how Ornstein and Uhlenbeck introduced this process in the 1930s as a more accurate model of Brownian motion than previous models. The process is defined as the unique solution to the stochastic differential equation dY(t) = -KY(t)dt + dX(t), where X(t) is a driving Lévy process and K is a matrix. Key properties are that Y(t) has càdlàg paths and is a Markov process. The process generalizes previous models where X(t) was Brownian motion.
What happens when the Kolmogorov-Zakharov spectrum is nonlocal?Colm Connaughton
This document summarizes research on the behavior of the Kolmogorov-Zakharov (KZ) spectrum when it is nonlocal. It examines a model of cluster-cluster aggregation described by the Smoluchowski equation, which can be viewed as a model of 3-wave turbulence without backscatter. The research finds that when the exponents in the interaction term satisfy certain conditions, the KZ spectrum is nonlocal. In this case, the stationary state has a novel functional form and can become unstable, leading to oscillatory behavior in the cascade dynamics at long times. Open questions remain about whether physical systems exhibit this behavior and how the results are affected by including backscatter terms.
kinks and cusps in the transition dynamics of a bloch statejiang-min zhang
We discuss the transition dynamics of a Bloch state in a 1D tight binding chain, under two scenarios, namely, weak periodical driving [1] and sudden quench [2]. In the former case, the survival probability of the initial Bloch state shows kinks periodically; it is a piece-wise linear function of time. In the latter, the survival probability (Loschmidt echo in this case) shows cusps periodically; it is a piece-wise quadratic function of time. Kinks in the former case are a perturbative effect, while cusps in the latter are a non-perturbative effect. The kinks and cusps are reminiscent of the so-called dynamical phase transtion termed by Heyl et al. [3].
[1] J. M. Zhang and M. Haque, Nonsmooth and level-resolved dynamics illustrated with the tight binding model, arXiv:1404.4280.
[2] J. M. Zhang and H. T. Yang, Cusps in the quenched dynamics of a bloch state, arXiv:1601.03569.
[3] M. Heyl, A. Polkovnikov, and S. Kehrein, Dynamical quantum phase transitions in the transverse-field Ising model, Phys. Rev. Lett. 110, 135704 (2013).
The document outlines auto-regressive (AR) processes of order p. It begins by introducing AR(p) processes formally and discussing white noise. It then derives the first and second moments of an AR(p) process. Specific details are provided about AR(1) and AR(2) processes, including equations for their variance as a function of the noise variance and AR coefficients. Examples of simulated AR(1) processes are shown for different coefficient values.
On estimating the integrated co volatility usingkkislas
This document proposes a method to estimate the integrated co-volatility of two asset prices using high-frequency data that contains both microstructure noise and jumps.
It considers two cases - when the jump processes of the two assets are independent, and when they are dependent. For the independent case, it proposes an estimator that is robust to jumps. For the dependent case, it proposes a threshold estimator that combines pre-averaging to remove noise with a threshold method to reduce the effect of jumps. It proves the estimators are consistent and establishes their central limit theorems. Simulation results are also presented to illustrate the performance of the proposed methods.
11.generalized and subset integrated autoregressive moving average bilinear t...Alexander Decker
This document proposes generalized integrated autoregressive moving average bilinear (GBL) time series models and subset generalized integrated autoregressive moving average bilinear (GSBL) models to achieve stationary for all nonlinear time series. It presents the models' formulations and discusses their properties including stationary, convergence, and parameter estimation. An algorithm is provided to fit the one-dimensional models. The generalized models are applied to Wolfer sunspot numbers and the GBL model is found to perform better than the GSBL model.
Neutral Electronic Excitations: a Many-body approach to the optical absorptio...Elena Cannuccia
This document summarizes the use of the Bethe-Salpeter equation to calculate optical absorption spectra by accounting for many-body effects. It describes how the Bethe-Salpeter equation is derived from the Kubo formula and accounts for electron-hole interactions through the screened Coulomb potential. The Bethe-Salpeter equation describes the propagation of coupled electron-hole pairs and their contribution to optical absorption. It is shown to reduce to the standard Bethe-Salpeter equation within the GW approximation.
This document discusses conditions for an infinite tuple of operators (T1, T2, ...) on a Banach space B to be syndetically hypercyclic. It defines what it means for an infinite tuple to be syndetically hypercyclic and provides two main results: 1) An infinite tuple satisfies the hypercyclicity criterion if and only if it is syndetically hypercyclic for any syndetic sequences. 2) An infinite tuple T is weakly mixing if and only if for any open subsets U and V of B and any syndetic sequences, there exist elements of the sequences such that T1T2...(U) intersects V. The document provides proofs of these results.
Omiros' talk on the Bernoulli factory problemBigMC
This document summarizes previous work on simulating events of unknown probability using reverse time martingales. It discusses von Neumann's solution to the Bernoulli factory problem where f(p)=1/2. It also summarizes the Keane-O'Brien existence result, the Nacu-Peres Bernstein polynomial approach, and issues with implementing the Nacu-Peres algorithm at large n due to the large number of strings involved. It proposes developing a reverse time martingale approach to address these issues.
The document discusses finite speed approximations to the Navier-Stokes equations. It introduces relaxation approximations and a damped wave equation approximation as two methods to derive finite speed approximations. It then discusses a vector BGK approximation, which takes a kinetic approach using a system of hyperbolic equations that approximates the Boltzmann equation and can be shown to converge to the incompressible Navier-Stokes equations in the diffusive limit. The document provides details on the vector BGK model, including compatibility conditions, conservation laws, and proofs of consistency, stability, and existence of global solutions.
This document describes unbiased Markov chain Monte Carlo (MCMC) methods using coupled Markov chains. It begins by discussing how standard MCMC estimators are biased due to initialization and finite simulation length. It then introduces the idea of running two coupled Markov chains such that they meet and become equal after some meeting time τ. The difference in function values between the chains can then be used to construct an unbiased estimator. Several methods for designing coupled chains that meet this criterion are described, including couplings of popular MCMC algorithms like Metropolis-Hastings. Conditions under which the resulting estimators are guaranteed to be unbiased and have good statistical properties are also outlined.
Couplings of Markov chains and the Poisson equation Pierre Jacob
The document discusses couplings of Markov chains and the Poisson equation. It begins with an outline introducing couplings as a technique to study Markov chain convergence rates. An example is provided of a Gibbs sampler motivated by Dempster-Shafer inference, known as the donkey walk. A common random numbers coupling of the donkey walk yields an explicit bound on the Wasserstein distance between the distribution after t steps and the stationary distribution.
The document discusses Approximate Bayesian Computation (ABC). ABC allows inference for statistical models where the likelihood function is not available in closed form. ABC works by simulating data under different parameter values and comparing simulated to observed data. ABC has been used for model choice by comparing evidence for different models. Consistency of ABC for model choice depends on the criterion used and asymptotic identifiability of the parameters.
NITheP WITS node Seminar by Dr Dr. Roland Cristopher F. Caballar (NITheP/UKZN)
TITLE: "One-Dimensional Homogeneous Open Quantum Walks"
ABSTRACT: In this talk, we consider a system undergoing an open quantum walk on a one-dimensional lattice. Each jump of the system between adjacent lattice points in a given direction corresponds to a jump operator, with these jump operators either commuting or not commuting. We examine the dynamics of the system undergoing this open quantum walk, in particular deriving analytically the probability distribution of the system, as well as examining numerically the behavior of the probability distribution over long time steps. The resulting distribution is shown to have multiple components, which fall under two general categories, namely normal and solitonic components. The analytic computation of the probability distribution for the system undergoing this open quantum walk allows us to determine at any instant of time the dynamical properties of the system.
Cluster-cluster aggregation with (complete) collisional fragmentationColm Connaughton
This document summarizes a presentation on cluster-cluster aggregation models with collisional fragmentation. It discusses mean-field theories of aggregation with a source of monomers and collision-induced fragmentation. Stationary solutions to the Smoluchowski equation are presented for both local and nonlocal aggregation kernels. While stationary nonlocal solutions exist, they are dynamically unstable. Simplified models with complete fragmentation and a source/sink of monomers produce exact solutions analogous to previous work. Nonlocality and the instability of stationary states require further study.
Nonequilibrium Statistical Mechanics of Cluster-cluster Aggregation Warwick ...Colm Connaughton
This document summarizes a talk on nonequilibrium statistical mechanics of cluster-cluster aggregation. The talk focused on theoretical models of particle clustering, including simple models where particles perform random walks and merge upon contact, and more sophisticated models that track the distribution of cluster sizes over time using the Smoluchowski equation. It discussed self-similar solutions and stationary solutions of the Smoluchowski equation. It also described the gelation transition that can occur when clusters absorb smaller clusters rapidly, violating the assumption of mass conservation and leading to clusters of infinite size.
A new approach to constants of the motion and the helmholtz conditionsAlexander Decker
1) The document discusses a new approach to determining constants of motion based on Helmholtz conditions for the existence of a Lagrangian.
2) It shows that constants of motion determined by this new approach vanish when the symmetry group is related to an actual invariance of the Lagrangian, as in this case the classical Noether invariants can be used.
3) The key equations of the approach are presented, including an equation (9) that provides an expression for constants of motion associated with any symmetry of a dynamical system with a Lagrangian.
The document provides an introduction to Brownian motion by starting with a one-dimensional discrete case modeled as a drunk walking randomly. It shows that Brownian motion has the properties of being memory-less, homogeneous in time and space. By taking the limit of discrete steps, the model arrives at continuous Brownian motion described by a partial differential equation. The document then briefly outlines the history of Brownian motion from its discovery to developments in modeling it as a stochastic process.
Unbiased Markov chain Monte Carlo methods Pierre Jacob
This document describes unbiased Markov chain Monte Carlo methods for approximating integrals with respect to a target probability distribution π. It introduces the idea of coupling two Markov chains such that their states are equal with positive probability, which can be used to construct an unbiased estimator of integrals of the form Eπ[h(X)]. The document outlines conditions under which the proposed estimator is unbiased and has finite variance. It also discusses implementations of coupled Markov chains for common MCMC algorithms like Metropolis-Hastings and Gibbs sampling.
This document summarizes research on modeling wildfires and turbulent premixed combustion. It discusses how turbulence affects wildfire propagation through turbulent transport of hot air, making the fire front position random. It presents a level set method for modeling deterministic and random fire fronts, accounting for turbulence. It also discusses using a Lagrangian approach to model turbulent premixed combustion, describing how the burned mass fraction evolves due to particle motion, flame front velocity, and curvature.
Nonlinear transport phenomena: models, method of solving and unusual features...SSA KPI
The document is a lecture on nonlinear transport phenomena and patterns. It discusses heat transport models described by balance equations. The key equations presented are the heat transport balance equation and the resulting partial differential equation in one spatial variable. The lecture aims to discuss patterns supported by various transport equations, defined as non-monotonic solutions that maintain shape or evolve self-similarly over time.
The document provides an overview of Session 1 of a COMSOL training series, which introduces COMSOL software. It discusses multiphysics simulation, which involves modeling multiple interacting physical phenomena, like fluid flow, heat transfer, and electrodynamics. It gives examples of why simulation is useful, such as design validation, optimization, and analysis. It then outlines the basic steps of simulation: defining physical phenomena with PDEs, discretizing the domain, solving the PDEs, and visualizing results. Finally, it previews the example simulation of a microfluidic mixer.
The document outlines auto-regressive (AR) processes of order p. It begins by introducing AR(p) processes formally and discussing white noise. It then derives the first and second moments of an AR(p) process. Specific details are provided about AR(1) and AR(2) processes, including equations for their variance as a function of the noise variance and AR coefficients. Examples of simulated AR(1) processes are shown for different coefficient values.
On estimating the integrated co volatility usingkkislas
This document proposes a method to estimate the integrated co-volatility of two asset prices using high-frequency data that contains both microstructure noise and jumps.
It considers two cases - when the jump processes of the two assets are independent, and when they are dependent. For the independent case, it proposes an estimator that is robust to jumps. For the dependent case, it proposes a threshold estimator that combines pre-averaging to remove noise with a threshold method to reduce the effect of jumps. It proves the estimators are consistent and establishes their central limit theorems. Simulation results are also presented to illustrate the performance of the proposed methods.
11.generalized and subset integrated autoregressive moving average bilinear t...Alexander Decker
This document proposes generalized integrated autoregressive moving average bilinear (GBL) time series models and subset generalized integrated autoregressive moving average bilinear (GSBL) models to achieve stationary for all nonlinear time series. It presents the models' formulations and discusses their properties including stationary, convergence, and parameter estimation. An algorithm is provided to fit the one-dimensional models. The generalized models are applied to Wolfer sunspot numbers and the GBL model is found to perform better than the GSBL model.
Neutral Electronic Excitations: a Many-body approach to the optical absorptio...Elena Cannuccia
This document summarizes the use of the Bethe-Salpeter equation to calculate optical absorption spectra by accounting for many-body effects. It describes how the Bethe-Salpeter equation is derived from the Kubo formula and accounts for electron-hole interactions through the screened Coulomb potential. The Bethe-Salpeter equation describes the propagation of coupled electron-hole pairs and their contribution to optical absorption. It is shown to reduce to the standard Bethe-Salpeter equation within the GW approximation.
This document discusses conditions for an infinite tuple of operators (T1, T2, ...) on a Banach space B to be syndetically hypercyclic. It defines what it means for an infinite tuple to be syndetically hypercyclic and provides two main results: 1) An infinite tuple satisfies the hypercyclicity criterion if and only if it is syndetically hypercyclic for any syndetic sequences. 2) An infinite tuple T is weakly mixing if and only if for any open subsets U and V of B and any syndetic sequences, there exist elements of the sequences such that T1T2...(U) intersects V. The document provides proofs of these results.
Omiros' talk on the Bernoulli factory problemBigMC
This document summarizes previous work on simulating events of unknown probability using reverse time martingales. It discusses von Neumann's solution to the Bernoulli factory problem where f(p)=1/2. It also summarizes the Keane-O'Brien existence result, the Nacu-Peres Bernstein polynomial approach, and issues with implementing the Nacu-Peres algorithm at large n due to the large number of strings involved. It proposes developing a reverse time martingale approach to address these issues.
The document discusses finite speed approximations to the Navier-Stokes equations. It introduces relaxation approximations and a damped wave equation approximation as two methods to derive finite speed approximations. It then discusses a vector BGK approximation, which takes a kinetic approach using a system of hyperbolic equations that approximates the Boltzmann equation and can be shown to converge to the incompressible Navier-Stokes equations in the diffusive limit. The document provides details on the vector BGK model, including compatibility conditions, conservation laws, and proofs of consistency, stability, and existence of global solutions.
This document describes unbiased Markov chain Monte Carlo (MCMC) methods using coupled Markov chains. It begins by discussing how standard MCMC estimators are biased due to initialization and finite simulation length. It then introduces the idea of running two coupled Markov chains such that they meet and become equal after some meeting time τ. The difference in function values between the chains can then be used to construct an unbiased estimator. Several methods for designing coupled chains that meet this criterion are described, including couplings of popular MCMC algorithms like Metropolis-Hastings. Conditions under which the resulting estimators are guaranteed to be unbiased and have good statistical properties are also outlined.
Couplings of Markov chains and the Poisson equation Pierre Jacob
The document discusses couplings of Markov chains and the Poisson equation. It begins with an outline introducing couplings as a technique to study Markov chain convergence rates. An example is provided of a Gibbs sampler motivated by Dempster-Shafer inference, known as the donkey walk. A common random numbers coupling of the donkey walk yields an explicit bound on the Wasserstein distance between the distribution after t steps and the stationary distribution.
The document discusses Approximate Bayesian Computation (ABC). ABC allows inference for statistical models where the likelihood function is not available in closed form. ABC works by simulating data under different parameter values and comparing simulated to observed data. ABC has been used for model choice by comparing evidence for different models. Consistency of ABC for model choice depends on the criterion used and asymptotic identifiability of the parameters.
NITheP WITS node Seminar by Dr Dr. Roland Cristopher F. Caballar (NITheP/UKZN)
TITLE: "One-Dimensional Homogeneous Open Quantum Walks"
ABSTRACT: In this talk, we consider a system undergoing an open quantum walk on a one-dimensional lattice. Each jump of the system between adjacent lattice points in a given direction corresponds to a jump operator, with these jump operators either commuting or not commuting. We examine the dynamics of the system undergoing this open quantum walk, in particular deriving analytically the probability distribution of the system, as well as examining numerically the behavior of the probability distribution over long time steps. The resulting distribution is shown to have multiple components, which fall under two general categories, namely normal and solitonic components. The analytic computation of the probability distribution for the system undergoing this open quantum walk allows us to determine at any instant of time the dynamical properties of the system.
Cluster-cluster aggregation with (complete) collisional fragmentationColm Connaughton
This document summarizes a presentation on cluster-cluster aggregation models with collisional fragmentation. It discusses mean-field theories of aggregation with a source of monomers and collision-induced fragmentation. Stationary solutions to the Smoluchowski equation are presented for both local and nonlocal aggregation kernels. While stationary nonlocal solutions exist, they are dynamically unstable. Simplified models with complete fragmentation and a source/sink of monomers produce exact solutions analogous to previous work. Nonlocality and the instability of stationary states require further study.
Nonequilibrium Statistical Mechanics of Cluster-cluster Aggregation Warwick ...Colm Connaughton
This document summarizes a talk on nonequilibrium statistical mechanics of cluster-cluster aggregation. The talk focused on theoretical models of particle clustering, including simple models where particles perform random walks and merge upon contact, and more sophisticated models that track the distribution of cluster sizes over time using the Smoluchowski equation. It discussed self-similar solutions and stationary solutions of the Smoluchowski equation. It also described the gelation transition that can occur when clusters absorb smaller clusters rapidly, violating the assumption of mass conservation and leading to clusters of infinite size.
A new approach to constants of the motion and the helmholtz conditionsAlexander Decker
1) The document discusses a new approach to determining constants of motion based on Helmholtz conditions for the existence of a Lagrangian.
2) It shows that constants of motion determined by this new approach vanish when the symmetry group is related to an actual invariance of the Lagrangian, as in this case the classical Noether invariants can be used.
3) The key equations of the approach are presented, including an equation (9) that provides an expression for constants of motion associated with any symmetry of a dynamical system with a Lagrangian.
The document provides an introduction to Brownian motion by starting with a one-dimensional discrete case modeled as a drunk walking randomly. It shows that Brownian motion has the properties of being memory-less, homogeneous in time and space. By taking the limit of discrete steps, the model arrives at continuous Brownian motion described by a partial differential equation. The document then briefly outlines the history of Brownian motion from its discovery to developments in modeling it as a stochastic process.
Unbiased Markov chain Monte Carlo methods Pierre Jacob
This document describes unbiased Markov chain Monte Carlo methods for approximating integrals with respect to a target probability distribution π. It introduces the idea of coupling two Markov chains such that their states are equal with positive probability, which can be used to construct an unbiased estimator of integrals of the form Eπ[h(X)]. The document outlines conditions under which the proposed estimator is unbiased and has finite variance. It also discusses implementations of coupled Markov chains for common MCMC algorithms like Metropolis-Hastings and Gibbs sampling.
This document summarizes research on modeling wildfires and turbulent premixed combustion. It discusses how turbulence affects wildfire propagation through turbulent transport of hot air, making the fire front position random. It presents a level set method for modeling deterministic and random fire fronts, accounting for turbulence. It also discusses using a Lagrangian approach to model turbulent premixed combustion, describing how the burned mass fraction evolves due to particle motion, flame front velocity, and curvature.
Nonlinear transport phenomena: models, method of solving and unusual features...SSA KPI
The document is a lecture on nonlinear transport phenomena and patterns. It discusses heat transport models described by balance equations. The key equations presented are the heat transport balance equation and the resulting partial differential equation in one spatial variable. The lecture aims to discuss patterns supported by various transport equations, defined as non-monotonic solutions that maintain shape or evolve self-similarly over time.
The document provides an overview of Session 1 of a COMSOL training series, which introduces COMSOL software. It discusses multiphysics simulation, which involves modeling multiple interacting physical phenomena, like fluid flow, heat transfer, and electrodynamics. It gives examples of why simulation is useful, such as design validation, optimization, and analysis. It then outlines the basic steps of simulation: defining physical phenomena with PDEs, discretizing the domain, solving the PDEs, and visualizing results. Finally, it previews the example simulation of a microfluidic mixer.
This slide deals with different aspects of Comsol Multiphysics and it's possibility in the future as multiple physics properties can be studied simultaneously with the help of different inbuilt or user-defined modules in this software.
This document discusses different types of hydraulic pressure control valves. It describes pressure relief valves, pilot operated relief valves, sequence control valves, and other types. Pressure relief valves limit pressure by diverting fluid to the reservoir when pressure reaches a set point. Pilot operated relief valves use a piston or spool controlled by a pilot valve. Sequence valves provide flow to a second actuator after the first reaches a threshold pressure. The document also provides examples of applications for different valve types.
This document discusses vapor-liquid equilibrium (VLE) calculations for various binary and ternary systems using the software HYSYS. It provides examples of calculating bubble point pressures, dew point pressures, and compositions for systems such as methanol/methyl acetate at different temperatures and compositions. It also assigns homework problems calculating VLE properties for systems like ethyl ethanoate/n-heptane and methane/ethylene/ethane using assumptions like Raoult's law.
This document provides an overview of using HYSYS simulation software to model and analyze chemical processes. It discusses setting up a HYSYS case by adding components, selecting a fluid package, and entering the simulation environment. It also covers defining process units like separators and heat exchangers, specifying stream properties, performing flash calculations, and generating workbooks. The document is intended as an introduction for students to learn the basic functionality of HYSYS through examples of common unit operations.
Simulation-Led Design Using SolidWorks® and COMSOL Multiphysics®Design World
Multiphysics has earned the reputation as an excellent approach for simulation in engineering and science. Applying multiphysics simulation early in the product development process brings you reliable computer models to verify and optimize your designs
This webinar will demonstrate how the COMSOL LiveLink for SolidWorks bridges the gap between design and analysis, integrating real-world simulation right into the CAD design environment of SolidWorks.
Attend this webinar to learn:
The importance of multiphysics modeling for true simulation of real-world applications
How to integrate analysis into the design process
the workflow of modeling with COMSOL Multiphysics and SolidWorks
Pressure Relief Valve Sizing for Single Phase FlowVikram Sharma
This presentation file provides a quick refresher to pressure relief valve sizing for single phase flow. The calculation guideline is as per API Std 520.
Thermodynamics deals with the effects of work, heat and energy on systems. It considers macroscopic and microscopic changes. The laws of thermodynamics are:
1) Zeroth law - If two systems are in thermal equilibrium with a third, they are in equilibrium with each other.
2) First law - The change in internal energy of a closed system equals the heat supplied minus the work done.
3) Second law - Heat cannot spontaneously flow from a cold body to a hot body.
4) Third law - The entropy of a system approaches a constant value as the temperature approaches absolute zero.
The document discusses different types of compressors used to increase air pressure. It describes reciprocating compressors which use pistons to compress air inside cylinders. Rotary compressors like screw, vane, and lobe compressors compress air using rotating elements. Centrifugal and axial compressors accelerate air to increase pressure, with centrifugal compressors using impellers and axial using rotating and stationary blades in stages. The document provides details on components and operating principles of these compressor types.
1. The document provides solutions to homework problems involving partial differential equations.
2. Problem 1 solves the wave equation utt = c2uxx using d'Alembert's formula to find the solution u(x,t).
3. Problem 2 proves that if the initial conditions φ and ψ are odd functions, then the solution u(x,t) is also an odd function.
This document analyzes conditions for a tuple of operators to be topologically mixing on a Fréchet space. It defines what it means for an operator tuple to be topologically mixing and hypercyclic. The main result is that if an operator tuple satisfies the hypercyclicity criterion for syndetic sequences, then it is topologically mixing. The hypercyclicity criterion and proof of the main theorem are discussed in detail. References analyzing hypercyclic operators, weighted shifts, and related topics are also provided.
This document summarizes research on quantum chaos, including the principle of uniform semiclassical condensation of Wigner functions, spectral statistics in mixed systems, and dynamical localization of chaotic eigenstates. It discusses how in the semiclassical limit, Wigner functions condense uniformly on classical invariant components. For mixed systems, the spectrum can be seen as a superposition of regular and chaotic level sequences. Localization effects can be observed if the Heisenberg time is shorter than the classical diffusion time. The document presents an analytical formula called BRB that describes the transition between Poisson and random matrix statistics. An example is given of applying this to analyze the level spacing distribution for a billiard system.
This document summarizes a basic science project on the applications of differential equations. It discusses how differentiation can be used to model population growth over time using exponential functions. As an example, it shows how to calculate the time needed for a population to triple if it is known to double every 30 years. The document concludes that differential equations have many applications in predicting real-world system behaviors over time.
This document discusses the derivation of a Quotient Rule Integration by Parts formula. It shows how the student Victor Reynolds asked if a similar formula could be derived from the Quotient Rule as the standard Integration by Parts formula is derived from the Product Rule. The author proceeds to derive such a Quotient Rule Integration by Parts formula. An example application of the new formula is also shown. However, the formula does not appear in calculus texts because it provides only a slight technical advantage over the standard formula and requires the same integral computations.
All of material inside is un-licence, kindly use it for educational only but please do not to commercialize it.
Based on 'ilman nafi'an, hopefully this file beneficially for you.
Thank you.
The lattice Boltzmann equation: background and boundary conditionsTim Reis
The document summarizes the lattice Boltzmann equation (LBE) method for computational fluid dynamics. It begins by discussing how the LBE is derived from kinetic theory and discrete kinetic theory, rather than directly from fluid equations. Taking moments of the discrete Boltzmann equation recovers the compressible Navier-Stokes equations through Chapman-Enskog expansion. The document then derives an analytic solution for the LBE governing Poiseuille flow between stationary walls under the influence of gravity.
This document discusses anomaly and parity odd transport coefficients in 1+1 dimensions. It begins by defining what an anomaly is, noting that a symmetry of classical physics may not hold at the quantum level. It then relates anomalies to hydrodynamics, showing how anomalies can constrain transport coefficients. The document evaluates the U(1) chiral anomaly using the Fujikawa method and relates it to hydrodynamic equations. It then uses the Kubo formula from linear response theory to evaluate the relevant current-stress tensor correlator from finite temperature field theory and performs Matsubara sums to obtain an expression for the parity odd transport coefficient.
1. Where distributions comes from?
2. Interpret and compare distributions.
3. Why normal, chi-square, t and F distributions?
4. Distributions for survivals.
This document discusses developing near-optimal state feedback controllers for nonlinear discrete-time systems using iterative approximate dynamic programming (ADP) algorithms. Specifically:
1) An infinite-horizon optimal state feedback controller is developed for discrete-time systems based on the dual heuristic programming (DHP) algorithm.
2) A new optimal control scheme is developed using the generalized DHP (GDHP) algorithm and a discounted cost functional.
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Nonlinear transport phenomena: models, method of solving and unusual features (2)
1. Nonlinear transport phenomena:
models, method of solving and unusual
features
Vsevolod Vladimirov
AGH University of Science and technology, Faculty of Applied
Mathematics
´
Krakow, August 10, 2010
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 1 / 29
2. Burgers equation
Consider the second law of Newton for viscous incompressible
fluid:
∂ ui ∂ ui 1∂P
+ uj j
+ = ν ∆ ui , i = 1, ...n, n = 1, 2 or 3,
∂t ∂x ρ ∂ xi
u(t, x) is the velocity field,
∂ j ∂
∂ t + u ∂ xj is the time ( substantial) derivative;
ρ is the constant density ;
P is the pressure ;
ν is the viscosity coefficient;
2
∆ = n ∂∂x2 is the Laplace operator.
i=1
i
For P = const, n = 1, we get the Burgers equation
ut + u ux = ν ux x . (1)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 2 / 29
3. Burgers equation
Consider the second law of Newton for viscous incompressible
fluid:
∂ ui ∂ ui 1∂P
+ uj j
+ = ν ∆ ui , i = 1, ...n, n = 1, 2 or 3,
∂t ∂x ρ ∂ xi
u(t, x) is the velocity field,
∂ j ∂
∂ t + u ∂ xj is the time ( substantial) derivative;
ρ is the constant density ;
P is the pressure ;
ν is the viscosity coefficient;
2
∆ = n ∂∂x2 is the Laplace operator.
i=1
i
For P = const, n = 1, we get the Burgers equation
ut + u ux = ν ux x . (1)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 2 / 29
4. Burgers equation
Consider the second law of Newton for viscous incompressible
fluid:
∂ ui ∂ ui 1∂P
+ uj j
+ = ν ∆ ui , i = 1, ...n, n = 1, 2 or 3,
∂t ∂x ρ ∂ xi
u(t, x) is the velocity field,
∂ j ∂
∂ t + u ∂ xj is the time ( substantial) derivative;
ρ is the constant density ;
P is the pressure ;
ν is the viscosity coefficient;
2
∆ = n ∂∂x2 is the Laplace operator.
i=1
i
For P = const, n = 1, we get the Burgers equation
ut + u ux = ν ux x . (1)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 2 / 29
5. Hyperbolic generalization to Burgers equation
Let us consider delayed equation
∂ u(t + τ, x)
+ u(t, x) ux (t, x) = ν ux x (t, x).
∂t
Applying to the term ∂ u(t+τ, x) the Taylor formula, we get, up
∂t
to O(τ 2 ) the equation called the hyperbolic generalization of
the Burgers equation (GBE to abbreviate):
τ utt + ut + u ux = ν ux x . (2)
GBE appears when modeling transport phenomena in media
possessing internal structure: granular media,polymers, cellular
structures in biology.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29
6. Hyperbolic generalization to Burgers equation
Let us consider delayed equation
∂ u(t + τ, x)
+ u(t, x) ux (t, x) = ν ux x (t, x).
∂t
Applying to the term ∂ u(t+τ, x) the Taylor formula, we get, up
∂t
to O(τ 2 ) the equation called the hyperbolic generalization of
the Burgers equation (GBE to abbreviate):
τ utt + ut + u ux = ν ux x . (2)
GBE appears when modeling transport phenomena in media
possessing internal structure: granular media,polymers, cellular
structures in biology.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29
7. Hyperbolic generalization to Burgers equation
Let us consider delayed equation
∂ u(t + τ, x)
+ u(t, x) ux (t, x) = ν ux x (t, x).
∂t
Applying to the term ∂ u(t+τ, x) the Taylor formula, we get, up
∂t
to O(τ 2 ) the equation called the hyperbolic generalization of
the Burgers equation (GBE to abbreviate):
τ utt + ut + u ux = ν ux x . (2)
GBE appears when modeling transport phenomena in media
possessing internal structure: granular media,polymers, cellular
structures in biology.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29
8. Hyperbolic generalization to Burgers equation
Let us consider delayed equation
∂ u(t + τ, x)
+ u(t, x) ux (t, x) = ν ux x (t, x).
∂t
Applying to the term ∂ u(t+τ, x) the Taylor formula, we get, up
∂t
to O(τ 2 ) the equation called the hyperbolic generalization of
the Burgers equation (GBE to abbreviate):
τ utt + ut + u ux = ν ux x . (2)
GBE appears when modeling transport phenomena in media
possessing internal structure: granular media,polymers, cellular
structures in biology.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29
9. Various generalizations of Burgers equation
Convection-reaction diffusion equation
ut + u ux = ν [un ux ]x + f (u), (3)
and its hyperbolic generalization
τ ut t + ut + u ux = ν [un ux ]x + f (u) (4)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 4 / 29
10. Various generalizations of Burgers equation
Convection-reaction diffusion equation
ut + u ux = ν [un ux ]x + f (u), (3)
and its hyperbolic generalization
τ ut t + ut + u ux = ν [un ux ]x + f (u) (4)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 4 / 29
11. Solution to BE
Lemma 1. BE is connected with the equation
1 2
ψt + ψx = ν ψ x x (5)
2
by means of the transformation
u2
ψx = u, ψt = ν ux − . (6)
2
Lemma 2. The equation (5) is connected with the heat
transport equation
Φt = ν Φx x
by means of the transformation
ψ = −2 ν log Φ.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 5 / 29
12. Solution to BE
Lemma 1. BE is connected with the equation
1 2
ψt + ψx = ν ψ x x (5)
2
by means of the transformation
u2
ψx = u, ψt = ν ux − . (6)
2
Lemma 2. The equation (5) is connected with the heat
transport equation
Φt = ν Φx x
by means of the transformation
ψ = −2 ν log Φ.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 5 / 29
13. Solution to BE
Lemma 1. BE is connected with the equation
1 2
ψt + ψx = ν ψ x x (5)
2
by means of the transformation
u2
ψx = u, ψt = ν ux − . (6)
2
Lemma 2. The equation (5) is connected with the heat
transport equation
Φt = ν Φx x
by means of the transformation
ψ = −2 ν log Φ.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 5 / 29
14. Corollary. Solution to the initial value problem
ut + u ux = ν u x x , (7)
u(0, x) = F (x)
is connected with the solution to the initial value problem
Φt = ν Φ x x , (8)
x
1
Φ(0, x) = exp − F (z) d z := θ(x)
2ν 0
via the transformation
u(t, x) = −2 ν {log[Φ(t, x)]}x .
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 6 / 29
15. Corollary. Solution to the initial value problem
ut + u ux = ν u x x , (7)
u(0, x) = F (x)
is connected with the solution to the initial value problem
Φt = ν Φ x x , (8)
x
1
Φ(0, x) = exp − F (z) d z := θ(x)
2ν 0
via the transformation
u(t, x) = −2 ν {log[Φ(t, x)]}x .
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 6 / 29
16. Corollary. Solution to the initial value problem
ut + u ux = ν u x x , (7)
u(0, x) = F (x)
is connected with the solution to the initial value problem
Φt = ν Φ x x , (8)
x
1
Φ(0, x) = exp − F (z) d z := θ(x)
2ν 0
via the transformation
u(t, x) = −2 ν {log[Φ(t, x)]}x .
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 6 / 29
17. Let us remind, that solution to the initial value problem (8) can
be presented by the formula
∞ (x−ξ)2
1
Φ(t, x) = √ θ(ξ) e− 4ν t d ξ.
4πν t −∞
Corollary. Solution to the initial value problem (7) is given by
the formula
∞ x−ξ − f (ξ;t, x)
−∞ t e dξ
2ν
u(t, x) = f (ξ;t, x)
, (9)
∞ −
−∞ e 2ν d ξ
where
ξ
(x − ξ)2
f (ξ; t, x) = F (z) d z +
0 2t
.
So, the formula (9)completely defines the solution to Cauchy
problem to BE!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29
18. Let us remind, that solution to the initial value problem (8) can
be presented by the formula
∞ (x−ξ)2
1
Φ(t, x) = √ θ(ξ) e− 4ν t d ξ.
4πν t −∞
Corollary. Solution to the initial value problem (7) is given by
the formula
∞ x−ξ − f (ξ;t, x)
−∞ t e dξ
2ν
u(t, x) = f (ξ;t, x)
, (9)
∞ −
−∞ e 2ν d ξ
where
ξ
(x − ξ)2
f (ξ; t, x) = F (z) d z +
0 2t
.
So, the formula (9)completely defines the solution to Cauchy
problem to BE!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29
19. Let us remind, that solution to the initial value problem (8) can
be presented by the formula
∞ (x−ξ)2
1
Φ(t, x) = √ θ(ξ) e− 4ν t d ξ.
4πν t −∞
Corollary. Solution to the initial value problem (7) is given by
the formula
∞ x−ξ − f (ξ;t, x)
−∞ t e dξ
2ν
u(t, x) = f (ξ;t, x)
, (9)
∞ −
−∞ e 2ν d ξ
where
ξ
(x − ξ)2
f (ξ; t, x) = F (z) d z +
0 2t
.
So, the formula (9)completely defines the solution to Cauchy
problem to BE!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29
20. Let us remind, that solution to the initial value problem (8) can
be presented by the formula
∞ (x−ξ)2
1
Φ(t, x) = √ θ(ξ) e− 4ν t d ξ.
4πν t −∞
Corollary. Solution to the initial value problem (7) is given by
the formula
∞ x−ξ − f (ξ;t, x)
−∞ t e dξ
2ν
u(t, x) = f (ξ;t, x)
, (9)
∞ −
−∞ e 2ν d ξ
where
ξ
(x − ξ)2
f (ξ; t, x) = F (z) d z +
0 2t
.
So, the formula (9)completely defines the solution to Cauchy
problem to BE!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29
21. Example: solution of the ”point explosion” problem
Let
u(0, x) = F (x) = Aδ(x)H(x),
1 (x−ξ)2 1 if x ≥ 0,
δ(x) = lim √ e− 4 ν t , H(x) = .
t→ +0 4πν t 0 if x < 0
Figure:
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 8 / 29
22. Example: solution of the ”point explosion” problem
Let
u(0, x) = F (x) = Aδ(x)H(x),
1 (x−ξ)2 1 if x ≥ 0,
δ(x) = lim √ e− 4 ν t , H(x) = .
t→ +0 4πν t 0 if x < 0
Figure:
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 8 / 29
23. Performing simple but tedious calculations, we finally get the
following solution to the point explosion problem:
x2
ν eR − 1 e− 4 ν t
u(t, x) = √ ,
t π x
(eR + 1) + erf( √4 ν t ) (1 − eR )
2
where z
2 2
erf(z) = √ e−x d x,
π 0
A
R= 2ν plays the role of the Reynolds number!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29
24. Performing simple but tedious calculations, we finally get the
following solution to the point explosion problem:
x2
ν eR − 1 e− 4 ν t
u(t, x) = √ ,
t π x
(eR + 1) + erf( √4 ν t ) (1 − eR )
2
where z
2 2
erf(z) = √ e−x d x,
π 0
A
R= 2ν plays the role of the Reynolds number!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29
25. Performing simple but tedious calculations, we finally get the
following solution to the point explosion problem:
x2
ν eR − 1 e− 4 ν t
u(t, x) = √ ,
t π x
(eR + 1) + erf( √4 ν t ) (1 − eR )
2
where z
2 2
erf(z) = √ e−x d x,
π 0
A
R= 2ν plays the role of the Reynolds number!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29
26. Performing simple but tedious calculations, we finally get the
following solution to the point explosion problem:
x2
ν eR − 1 e− 4 ν t
u(t, x) = √ ,
t π x
(eR + 1) + erf( √4 ν t ) (1 − eR )
2
where z
2 2
erf(z) = √ e−x d x,
π 0
A
R= 2ν plays the role of the Reynolds number!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29
27. Suppose now, that ν becomes very large. Then
x
R→ 0 eR ≈ 1 + R, erf √ ≈ 0,
4ν t
and
x2
ν A
e− 4 ν t A x2
u(t, x) = 2ν
√ + O(R2 ) ≈ √ e− 4 ν t .
t π 4πν t
Corollary.Solution to the ”point explosion” problem for the BE
approaches solution to the ”heat explosion” problem for the
linear heat transport equation, when ν becomes large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29
28. Suppose now, that ν becomes very large. Then
x
R→ 0 eR ≈ 1 + R, erf √ ≈ 0,
4ν t
and
x2
ν A
e− 4 ν t A x2
u(t, x) = 2ν
√ + O(R2 ) ≈ √ e− 4 ν t .
t π 4πν t
Corollary.Solution to the ”point explosion” problem for the BE
approaches solution to the ”heat explosion” problem for the
linear heat transport equation, when ν becomes large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29
29. Suppose now, that ν becomes very large. Then
x
R→ 0 eR ≈ 1 + R, erf √ ≈ 0,
4ν t
and
x2
ν A
e− 4 ν t A x2
u(t, x) = 2ν
√ + O(R2 ) ≈ √ e− 4 ν t .
t π 4πν t
Corollary.Solution to the ”point explosion” problem for the BE
approaches solution to the ”heat explosion” problem for the
linear heat transport equation, when ν becomes large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29
30. Suppose now, that ν becomes very large. Then
x
R→ 0 eR ≈ 1 + R, erf √ ≈ 0,
4ν t
and
x2
ν A
e− 4 ν t A x2
u(t, x) = 2ν
√ + O(R2 ) ≈ √ e− 4 ν t .
t π 4πν t
Corollary.Solution to the ”point explosion” problem for the BE
approaches solution to the ”heat explosion” problem for the
linear heat transport equation, when ν becomes large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29
31. For large R the way of getting the approximating formula is less clear, so
we restore to the results of the numerical simulation. Below it is shown the
solution to ”point explosion” problem obtained for ν = 0.05 and R = 35:
Figure:
It reminds the shock wave profile
x √
t
if t > 0, 0 < x < 2√ t,
A
u(t, x) = ,
0 if t > 0, x < 0 or x > 2 A t
which the BE ”shares” with the hyperbolic-type equation
ut + u ux = 0,
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 11 / 29
32. For large R the way of getting the approximating formula is less clear, so
we restore to the results of the numerical simulation. Below it is shown the
solution to ”point explosion” problem obtained for ν = 0.05 and R = 35:
Figure:
It reminds the shock wave profile
x √
t
if t > 0, 0 < x < 2√ t,
A
u(t, x) = ,
0 if t > 0, x < 0 or x > 2 A t
which the BE ”shares” with the hyperbolic-type equation
ut + u ux = 0,
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 11 / 29
33. For large R the way of getting the approximating formula is less clear, so
we restore to the results of the numerical simulation. Below it is shown the
solution to ”point explosion” problem obtained for ν = 0.05 and R = 35:
Figure:
It reminds the shock wave profile
x √
t
if t > 0, 0 < x < 2√ t,
A
u(t, x) = ,
0 if t > 0, x < 0 or x > 2 A t
which the BE ”shares” with the hyperbolic-type equation
ut + u ux = 0,
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 11 / 29
34. Figure:
A common solution
x
√
t if t > 0, 0 < x < 2√ t,
A
u(t, x) =
0 if t > 0, x < 0 or x > 2 A t,
to the Burgers and the Euler equations
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 12 / 29
35. So the solutions to the point explosion problem for BE are
completely different in the limiting cases: when
R = A/(2 ν) → 0 it coincides with the solution of the heat
explosion problem,
while for large R it reminds the shock wave solution!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 13 / 29
36. So the solutions to the point explosion problem for BE are
completely different in the limiting cases: when
R = A/(2 ν) → 0 it coincides with the solution of the heat
explosion problem,
while for large R it reminds the shock wave solution!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 13 / 29
37. The hyperbolic generalization of BE
Let us consider the Cauchy problem for the hyperbolic
generalization of BE:
τ utt + ut + u ux = ν ux x , (10)
u(0, x) = ϕ(x).
Considering the linearization of (10)
τ utt + ut + u0 ux = ν ux x ,
we can conclude, that the parameter C = ν/τ is equal to the
velocity of small (acoustic) perturbations.
If the initial perturbation ϕ(x) is a smooth compactly supported
function, and D = max ϕ(x), then the number M = D/C (the
”Mach number”) characterizes the evolution of nonlinear wave.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 14 / 29
38. The hyperbolic generalization of BE
Let us consider the Cauchy problem for the hyperbolic
generalization of BE:
τ utt + ut + u ux = ν ux x , (10)
u(0, x) = ϕ(x).
Considering the linearization of (10)
τ utt + ut + u0 ux = ν ux x ,
we can conclude, that the parameter C = ν/τ is equal to the
velocity of small (acoustic) perturbations.
If the initial perturbation ϕ(x) is a smooth compactly supported
function, and D = max ϕ(x), then the number M = D/C (the
”Mach number”) characterizes the evolution of nonlinear wave.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 14 / 29
39. The hyperbolic generalization of BE
Let us consider the Cauchy problem for the hyperbolic
generalization of BE:
τ utt + ut + u ux = ν ux x , (10)
u(0, x) = ϕ(x).
Considering the linearization of (10)
τ utt + ut + u0 ux = ν ux x ,
we can conclude, that the parameter C = ν/τ is equal to the
velocity of small (acoustic) perturbations.
If the initial perturbation ϕ(x) is a smooth compactly supported
function, and D = max ϕ(x), then the number M = D/C (the
”Mach number”) characterizes the evolution of nonlinear wave.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 14 / 29
40. Results of the numerical simulation: M = 0.3
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 15 / 29
41. Figure: M = 0.3
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 16 / 29
42. Figure: M = 0.3
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 17 / 29
43. Figure: M = 0.3
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 18 / 29
44. Figure: M = 0.3
The solution of the initial perturbation reminds the evolution of
the point explosion problem for BE in the case when
R = A/(2 ν) is large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 19 / 29
45. Results of the numerical simulation: M = 1.45
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 20 / 29
46. Figure: M = 1.45
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 21 / 29
47. Figure: M = 1.45
For M = 1 + ε a formation of the blow-up regime is observed at
the beginning of evolution,
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 22 / 29
48. Figure: M = 1.45
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 23 / 29
49. Figure: M = 1.45
but for larger t it is suppressed by viscosity and returns to the
shape of the BE solution!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 24 / 29
50. Results of the numerical simulation: M = 1.8
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 25 / 29
51. Figure: M = 1.8
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 26 / 29
52. Figure: M = 1.8
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 27 / 29
53. Figure: M = 1.8
For M = 1.8 (and larger ones) a blow-up regime is formed at
the wave front in finite time!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 28 / 29
54. Appendix 1. Calculation of point explosion problem
for BE
Since,
ξ ∞
−A, if ξ < 0,
F (x) d x = −A lim δ(x) φB (x) H(x) d x =
0+ B→+0 −∞ 0, if ξ > 0,
∞
where φB (x) is any C0 function such that φ(x)|<B, ξ> ≡ 1, and
supp φ ⊂ < B/2, ξ + B/2 > then
(x−ξ)2
2t − A if ξ < 0,
f (ξ; t, x) = (x−ξ)2
2 t , if ξ > 0.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 29 / 29
55. Appendix 1. Calculation of point explosion problem
for BE
Since,
ξ ∞
−A, if ξ < 0,
F (x) d x = −A lim δ(x) φB (x) H(x) d x =
0+ B→+0 −∞ 0, if ξ > 0,
∞
where φB (x) is any C0 function such that φ(x)|<B, ξ> ≡ 1, and
supp φ ⊂ < B/2, ξ + B/2 > then
(x−ξ)2
2t − A if ξ < 0,
f (ξ; t, x) = (x−ξ)2
2 t , if ξ > 0.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 29 / 29