This document presents a numerical method for solving nonlinear evolution equations of the form ut - Δf(u) = 0. The method involves solving a corresponding linear problem and performing simple algebraic operations at each time step. It is proven that the method is stable and convergent. Numerical experiments demonstrating the method are also presented. The method generalizes an existing technique for solving the Stefan problem and is related to another existing method. The document provides details on implementing the method to solve a specific nonlinear evolution equation with Dirichlet boundary conditions.
This document presents a theorem about the almost Norlund summability of conjugate Fourier series. It generalizes previous results by Pati (1961) and Singh and Singh (1993). The main theorem states that if the conjugate partial sums of a Fourier series satisfy certain conditions, including being bounded by a function that approaches 0 as n approaches infinity, then the conjugate Fourier series is almost Norlund summable to the integral of the function at every point where the integral exists. The proof utilizes lemmas about the behavior of the conjugate partial sums and applies mean value theorems to show the necessary conditions are met. References to previous related works are also provided.
This document summarizes the key results of a paper on parabolic Itô equations. A parabolic Itô equation is an equation of the form (du/dt) = Lu + f(u) + a, where L is a negative-definite operator, f is Lipschitz, and a is white noise. The paper proves:
I. There exists a unique non-anticipating solution u with sup|u|2 < ∞.
II. The solution can be written as u = R + V, where R is stationary and limE|V|2 = 0.
III. If additional conditions hold, an explicit expression is found for the stationary density of R.
Average Polynomial Time Complexity Of Some NP-Complete ProblemsAudrey Britton
This document discusses the average polynomial time complexity of some NP-complete problems. Specifically:
- It proves that when N(n) = [n^e] for 0 < e < 2, the CLIQUEN(n) problem (finding a clique of size k in a graph with n vertices and at most N(n) edges) is NP-complete, yet can be solved in average and almost everywhere polynomial time.
- It considers the CLIQUE problem under different non-uniform distributions on the input graphs, showing the problem can be solved in average polynomial time for certain values of the distribution parameter p.
- The main result shows that a family of subproblems of CLIQUE that remain
Fixed point result in probabilistic metric spaceAlexander Decker
This document presents a fixed point theorem for four self-mappings in a probabilistic metric space. It begins with definitions related to probabilistic metric spaces, including Menger spaces. It then proves a common fixed point theorem for four self-mappings defined on a complete Menger space, where the mappings satisfy certain conditions. Specifically, the mappings have nested range properties and some pairs are weakly or semi-compatible. Additionally, there is a contraction condition involving a rational function. The theorem guarantees the existence and uniqueness of a common fixed point for the four mappings. An example is also provided to support the theorem.
Reciprocity Law For Flat Conformal Metrics With Conical SingularitiesLukasz Obara
The document is a thesis that establishes an analogue of Weil's reciprocity law for flat conformal metrics with conical singularities on Riemann surfaces. It introduces the necessary mathematical background, including definitions of isothermal coordinates and metrics with conical singularities. The main result proves a relationship between three flat conformally equivalent metrics, each with different conical singularities.
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
This document discusses various applications of graph theory to other areas of mathematics and other fields. It provides new graph theoretical proofs of Fermat's Little Theorem and the Nielsen-Schreier Theorem. It also discusses applications to problems in DNA sequencing, computer network security, scheduling, map coloring, and mobile phone networks. Specific algorithms for finding minimum vertex covers, vertex colorings, and matchings in graphs are applied to solve problems in these areas.
This document presents a theorem about the almost Norlund summability of conjugate Fourier series. It generalizes previous results by Pati (1961) and Singh and Singh (1993). The main theorem states that if the conjugate partial sums of a Fourier series satisfy certain conditions, including being bounded by a function that approaches 0 as n approaches infinity, then the conjugate Fourier series is almost Norlund summable to the integral of the function at every point where the integral exists. The proof utilizes lemmas about the behavior of the conjugate partial sums and applies mean value theorems to show the necessary conditions are met. References to previous related works are also provided.
This document summarizes the key results of a paper on parabolic Itô equations. A parabolic Itô equation is an equation of the form (du/dt) = Lu + f(u) + a, where L is a negative-definite operator, f is Lipschitz, and a is white noise. The paper proves:
I. There exists a unique non-anticipating solution u with sup|u|2 < ∞.
II. The solution can be written as u = R + V, where R is stationary and limE|V|2 = 0.
III. If additional conditions hold, an explicit expression is found for the stationary density of R.
Average Polynomial Time Complexity Of Some NP-Complete ProblemsAudrey Britton
This document discusses the average polynomial time complexity of some NP-complete problems. Specifically:
- It proves that when N(n) = [n^e] for 0 < e < 2, the CLIQUEN(n) problem (finding a clique of size k in a graph with n vertices and at most N(n) edges) is NP-complete, yet can be solved in average and almost everywhere polynomial time.
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Fixed point result in probabilistic metric spaceAlexander Decker
This document presents a fixed point theorem for four self-mappings in a probabilistic metric space. It begins with definitions related to probabilistic metric spaces, including Menger spaces. It then proves a common fixed point theorem for four self-mappings defined on a complete Menger space, where the mappings satisfy certain conditions. Specifically, the mappings have nested range properties and some pairs are weakly or semi-compatible. Additionally, there is a contraction condition involving a rational function. The theorem guarantees the existence and uniqueness of a common fixed point for the four mappings. An example is also provided to support the theorem.
Reciprocity Law For Flat Conformal Metrics With Conical SingularitiesLukasz Obara
The document is a thesis that establishes an analogue of Weil's reciprocity law for flat conformal metrics with conical singularities on Riemann surfaces. It introduces the necessary mathematical background, including definitions of isothermal coordinates and metrics with conical singularities. The main result proves a relationship between three flat conformally equivalent metrics, each with different conical singularities.
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
This document discusses various applications of graph theory to other areas of mathematics and other fields. It provides new graph theoretical proofs of Fermat's Little Theorem and the Nielsen-Schreier Theorem. It also discusses applications to problems in DNA sequencing, computer network security, scheduling, map coloring, and mobile phone networks. Specific algorithms for finding minimum vertex covers, vertex colorings, and matchings in graphs are applied to solve problems in these areas.
Iterative procedure for uniform continuous mapping.Alexander Decker
This document presents an iterative procedure for finding a common fixed point of a finite family of self-maps on a nonempty closed convex subset of a normed linear space. Specifically:
1. It defines an m-step iterative process that generates a sequence from an initial point by applying m self-maps from the family sequentially at each step.
2. It proves that if one of the maps is uniformly continuous and hemicontractive, with bounded range, and the family has a nonempty common fixed point set, then the iterative sequence converges strongly to a common fixed point.
3. It extends previous results by allowing some maps in the family to satisfy only asymptotic conditions, rather than uniform continuity. The conditions
This document presents a study investigating the effects of variable thermal conductivity, heat source/sink, and variable free stream on fluid flow and heat transfer over a stretching sheet near a stagnation point. The governing equations are transformed using similarity transformations and solved numerically using the Homotopy Perturbation Method. The velocity and temperature profiles are presented in graphs showing the effects of varying thermal conductivity and other parameters. Skin friction coefficient and Nusselt number are also derived and their numerical values in tables show their relationships with physical parameters.
Fixed point result in menger space with ea propertyAlexander Decker
This document presents a fixed point theorem for four self-maps in a Menger space. It begins by defining key concepts related to Menger spaces including probabilistic metric spaces, t-norms, neighborhoods, convergence, Cauchy sequences, and completeness. It then introduces properties like weakly compatible maps, property (EA), and JSR mappings. The main result, Theorem 3.1, proves the existence of a common fixed point for four self-maps under conditions that the map pairs satisfy a common property (EA) and are closed, JSR mappings satisfying an inequality involving the probabilistic distance functions.
This document summarizes the topological structure of ternary residue curve maps, which describe the dynamics of ternary distillation processes. It introduces differential equations that model ternary distillation and place a meaningful structure on ternary phase diagrams. By recognizing this structure is subject to the Poincaré-Hopf index theorem, the authors obtained a topological relationship between azeotropes and pure components in ternary mixtures. This relationship provides useful information about ternary mixture distillation behavior and predicts situations where ternary azeotropes cannot occur.
1. The document discusses the Fourier transform and how it can be used to represent a signal as the sum of many harmonic signals with different frequencies and amplitudes.
2. It introduces the ricker wavelet function and explains how a ricker source signal can be decomposed into individual sinusoidal waves using a Fourier transform.
3. The document then discusses solving the 2D homogeneous acoustic wave equation using a Fourier transform approach. It shows that the Fourier transform of the wavefield solution is simply the Fourier transform of the source function multiplied by a Green's function.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Possible methods of providing further (and perhaps better) alternative solutions for the exponential
integral of aquifer parameter evaluation are investigated. Three known mathematical methods of approach
(comprising self-similar, separable variable and travelling wave) are applied, providing three relevant solutions.
Further analysis of the self-similar solution reveals that this provides an alternative solution involving normal
graph of drawdown versus the measurement intervals. The geomathematical relevance of this method is assessed
using data from aquifers from two chronologically different hydrogeological units – the Ajalli Sandstone and
Ogwashi-Asaba Formation. The results indicate good functional relationship with satisfactory transmissivity
values
This document provides a critical overview of equations used to analyze pumping test data to determine aquifer parameters. It discusses the assumptions and limitations of the commonly used Theis equation. Three alternative mathematical solutions to the exponential integral in the Theis equation are derived: self-similar, separable variable, and travelling wave. Further analysis of the self-similar solution provides an alternative approach using a normal graph of drawdown versus time intervals in log cycles to determine transmissivity. This method is applied to pumping test data from two aquifers in Nigeria, yielding transmissivity values that align with known properties of the aquifers. The analysis suggests the normal graph approach provides a good alternative or supplementary method to traditional curve-matching techniques for
1. The document provides an introduction to Fourier analysis and Fourier series. It discusses how periodic functions can be represented as the sum of infinite trigonometric terms.
2. Examples are given of arbitrary functions being approximated by Fourier series of increasing lengths. As the length of the series increases, the ability to mimic the behavior of the original function also increases.
3. The Fourier transform is introduced as a method to represent functions in terms of sine and cosine terms. It allows problems involving differential equations to be transformed into an algebraic form and then transformed back to find the solution.
Representation formula for traffic flow estimation on a networkGuillaume Costeseque
This document discusses representation formulas for traffic flow estimation on networks using Hamilton-Jacobi equations. It begins by motivating the use of HJ equations, noting advantages like smooth solutions and physically meaningful quantities. It then presents the basic ideas of Lax-Hopf formulas for solving HJ equations on networks, including a simple case study of a junction. The document outlines its topics which include notations from traffic flow modeling, basic recalls on Lax-Hopf formulas, HJ equations on networks, and a new approach.
This document discusses the convexity of the set of k-admissible functions on a compact Kähler manifold. It begins by introducing k-admissible functions and some necessary convex analysis concepts. It then proves four main results: 1) the log of elementary symmetric functions of eigenvalues is a convex function, 2) the set of matrices with eigenvalues in a convex set is convex, 3) certain functions of eigenvalues are convex, and 4) the set of k-admissible functions is convex. It uses results on conjugation of spectral functions from convex analysis to prove these results. The proofs rely on properties of convex, lower semicontinuous functions and indicator functions.
Spectral Continuity: (p, r) - Α P And (p, k) - QIOSR Journals
This document discusses spectral continuity properties for operators belonging to the classes of (p,r)-ΑP (absolute (p,r)-paranormal) operators and (p,k)-Q (quasihyponormal) operators. It is shown that if a sequence of operators from one of these classes converges in norm to an operator T in the same class, then the spectrum, Weyl spectrum, Browder spectrum, and essential surjectivity spectrum are continuous at T. Some key properties used in the proofs are that for these operator classes, the ascent is finite, the single valued extension property is satisfied, and the adjoint satisfies a version of Weyl's theorem.
On New Root Finding Algorithms for Solving Nonlinear Transcendental EquationsAI Publications
In this paper, we present new iterative algorithms to find a root of the given nonlinear transcendental equations. In the proposed algorithms, we use nonlinear Taylor’s polynomial interpolation and a modified error correction term with a fixed-point concept. We also investigated for possible extension of the higher order iterative algorithms in single variable to higher dimension. Several numerical examples are presented to illustrate the proposed algorithms.
We provide a comprehensive convergence analysis of the asymptotic preserving implicit-explicit particle-in-cell (IMEX-PIC) methods for the Vlasov–Poisson system with a strong magnetic field. This study is of utmost importance for understanding the behavior of plasmas in magnetic fusion devices such as tokamaks, where such a large magnetic field needs to be applied in order to keep the plasma particles on desired tracks.
This document provides exercises on Hawking radiation using scalar field theory in the Kruskal spacetime. It asks the student to find the radial equation for the scalar field and show that near the horizon, the field takes the form of ingoing and outgoing waves that are analytic in different coordinate systems. The document then derives the Klein-Gordon equation in Schwarzschild coordinates and uses separation of variables to obtain approximate solutions near the horizon. It shows that ingoing waves are regular on the future horizon but outgoing waves are not, and vice versa for the past horizon.
This document discusses linear response theory and time-dependent density functional theory (TDDFT) for calculating absorption spectroscopy. It begins by motivating the use of absorption spectroscopy to study many-body effects. It then outlines how to calculate the response of a system to a perturbation within linear response theory and the Kubo formula. The document discusses using TDDFT to include electron correlation effects beyond the independent particle and time-dependent Hartree approximations. It emphasizes that TDDFT provides an exact framework for calculating neutral excitations if the correct exchange-correlation functional is used.
This document contains information about a calculus project completed by students of the Mechanical Engineering department at Laxmi Institute of Technology in Sarigam. It includes the names and student IDs of 13 students who participated in the project. The document covers topics in multiple integrals, including double integrals, Fubini's theorem, double integrals in polar coordinates, and triple integrals. Formulas and examples are provided for each topic.
This document discusses the existence and uniqueness of renormalized solutions to a nonlinear multivalued elliptic problem with homogeneous Neumann boundary conditions and L1 data. Specifically, it considers the problem β(u) - div a(x, Du) ∋ f in Ω, with a(x, Du).η = 0 on ∂Ω, where f is an L1 function. It provides definitions of renormalized solutions and entropy solutions. The main result is the existence and uniqueness of renormalized solutions to this problem, which is proved using a priori estimates and a compactness argument with doubling of variables.
This document discusses the existence and uniqueness of renormalized solutions to a nonlinear multivalued elliptic problem with homogeneous Neumann boundary conditions and L1 data. Specifically, it considers the problem β(u) - div a(x, Du) ∋ f in Ω, with a(x, Du).η = 0 on ∂Ω. It defines renormalized solutions and entropy solutions for this problem. The main result is that under certain assumptions on the data, there exists a unique renormalized solution to the problem. The proof uses approximate methods, showing existence and uniqueness for a penalized approximation problem, and passing to the limit.
This document presents a temporal stability analysis of a swirling gas jet discharging into an ambient gas. The analysis considers the inviscid limit of the compressible swirling jet flow. The dispersion relation parameters studied include swirl number, Mach number, coflow velocity, and molar weight ratio. The base flow is solved using a self-similar solution, and a pseudospectral method is used to discretize the governing equations. The stability analysis derives the eigenvalue problem from the linearized Navier-Stokes equations to determine how perturbations of the base flow will grow or decay over time.
8 Pcs Vintage Lotus Letter Paper Stationery Writing PKim Daniels
The Articles of Confederation established the first government of the United States and unified the 13
original states as a confederation. It allowed the states to work together during the Revolutionary War
by giving certain powers to the Continental Congress. The Articles also defined state boundaries and
sovereignty, establishing the framework for how the new nation would be governed until it was
replaced by the U.S. Constitution in 1789.
Essay Writing Words 100 Useful Words And PhraseKim Daniels
The document discusses how Anzac Day became a sacred holiday in 1921 to commemorate and honor New Zealanders who fought in World War I. Anzac Day is celebrated annually on April 25th to remember those killed in the Gallipoli landing campaign and to honor returned service members. The day has become an important tradition in New Zealand to reflect on the sacrifices made during the war.
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1. The document discusses the Fourier transform and how it can be used to represent a signal as the sum of many harmonic signals with different frequencies and amplitudes.
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IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Possible methods of providing further (and perhaps better) alternative solutions for the exponential
integral of aquifer parameter evaluation are investigated. Three known mathematical methods of approach
(comprising self-similar, separable variable and travelling wave) are applied, providing three relevant solutions.
Further analysis of the self-similar solution reveals that this provides an alternative solution involving normal
graph of drawdown versus the measurement intervals. The geomathematical relevance of this method is assessed
using data from aquifers from two chronologically different hydrogeological units – the Ajalli Sandstone and
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values
This document provides a critical overview of equations used to analyze pumping test data to determine aquifer parameters. It discusses the assumptions and limitations of the commonly used Theis equation. Three alternative mathematical solutions to the exponential integral in the Theis equation are derived: self-similar, separable variable, and travelling wave. Further analysis of the self-similar solution provides an alternative approach using a normal graph of drawdown versus time intervals in log cycles to determine transmissivity. This method is applied to pumping test data from two aquifers in Nigeria, yielding transmissivity values that align with known properties of the aquifers. The analysis suggests the normal graph approach provides a good alternative or supplementary method to traditional curve-matching techniques for
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The film Born into Brothels follows British filmmakers documenting the lives of children living in the Red Light District of Calcutta, India. The filmmakers give the children cameras to take photos, hoping to provide them perspective and possibly change their futures. Through this, the film highlights the difficult circumstances faced by the children of sex workers but also their resilience and humanity.
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Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
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How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
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বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
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Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
3. 298 A. E. BERGER, H. BRÉZIS, J. C. W. ROGERS
decreasing function which is Lipschitz continuous on every bounded interval
and such that ƒ (0)= 0. Let L : D f I J c I 1
(fi) -• L1
(Q) be an unbounded linear
operator on L1
(Q) which satisfies the following conditions
L is a closed operator with dense domain D(L) in L1
(Q); for every X> 0,1 + X L
maps D (L ) one to one onto L1
(fi) and (I + XL)~ x
is a contraction in L1
(Q).In
other words, —L générâtes a linear contraction semigroup in L1
(Q) denoted by
S(t). (1)
For any X>0 and cpeL1
(Q),
sup (/ + XL)~x
<p ^ max {0, sup <p} (2)
Q n
[by sup we mean the essential supremum; if sup cp = oo, assumption (2) is empty].
There exists an a > 0 such that
alIttlIx^IlLulli for ail ueD(L) (3)
[throughout this paper, the LP
{Q) norm of any function cp will be denoted by
IMIJ-
Assumptions (l)-(3) are, for example, satisfied by Lu= —Au,
D(L)= {u € Wofl
(^); L u e L1
(Q)}(where L u is understood in the distribution
sensé). More generally we may take
where aijt eiieC1
^), aeL™(Q), both a and (a + ^ôai/dxj) are nonnegative
almost everywhere on Q, and for some positive constant a,
X<iyÇ.*Çj^a|£|2
a.e. in Q, for each ÇeRN
(see, e.g., theorem 8 of Brezis and Strauss [11]).
We are concerned with solving the évolution équation
^+Z,/(W ) = 0, U(0) = MO. (4)
The nonlinear operator Au = Lf(u) defined as an operator in //(fi) with
domain D(A)= {ueL1
(Q);ƒ (u)eD(L)} is m-accretive in L1
(Q),i.e., for every
R.A.I.R.O. Analyse numérique/Numerical Analysis
4. METHOD FORSOLVING THE PROBLEM Ut - A / ( u ) = 0 2 9 9
q>eLl
(Q) andevery X,>0, there is a unique solution ueD(A) of the équation
and in addition, themapping cp-> uis a contraction in L1
(Q) (see for example
theorem 1ofBrezis andStrauss [11]). Onthe other hand, D (A) isdense inL1
(Q)
(note for example that if q
> eL°°(Q), then %=(I + X A)~1
cp satisfies
IIM
x II2^ ||<
P II2 anc
* so ux-• q
>in L2
(Q) as A
, ->0). It follows that
Sg(t)u0=
is a contraction semigroup onD(A)=L1
(Q); Sg(i)u0 isthegeneralized solution
of (4) in thesense of Crandall-Liggett andBenilan (see [4, 10, 12,13, 14]).
The classical Stefan problem with homogeneous Dirichlet data can be cast
into theform (4)as described below. Consider theStefan problem posed interms
of température (v), with thefreezing point denoted byz, andfor convenience,
with constant material properties (c„cwt kif kw);
CiVt =kiAv for pel(t)~ {peQ; v(p,t)<z], £>0, (5a)
c^v^k^Av for peW{t)={peav(p,t)>z}, t>0, {5b)
v(p,t) =g(p,t) for peT, t>0, (5c)
v(p, r= 0)= wo(p) for peÖ. (5d)
The proper energy balance conditions on themoving interface
M(t)={peQ;v(p,t) =z}
is
-iQvï+kiV-^XVv for peM{t), t>0. (5e)
Here vistheunit normal to M{t)pointing into W{t),Xisthelatent heat of the
change of phase, Vv is the velocity of the interface in the direction of v, and
Vy(p, t)dénotes thelimit of v at p approached from within W{t), etc. Let
Ciîoïv<z ( fOforu<z
for u> z
Let theenthalpy w(p,t)= u(u(p, £))be defined by
u(v)~ c{Qdli-hXr(v)--rl (rt is an arbitrary constant). (6)
vol. 13, n° 4, 1979
5. 300 A. E. BERGER, H. BRÉZIS, J. C. W. ROGERS
Then when g — 0, (5) can be written in the form (4) with L = — À and (see [9, 15]) :
for ur£ru
(7)
f(u)= z foi r^u^r
z foi r^u^r^X,
z + kw(u — X— r1)/cw for u^rt +X.
To obtain zero Dirichlet data for (4), the constants rx and z are chosen so that
M(0) = 0 and/(0) = 0.
Problem (4) with L = - A and ƒ (u) = |u |a
"* u(a > 1) occurs in a model of gas
diffusion through a porous medium (hère u corresponds to gas density and
gradjii^"1
to velocity, see e.g. [1, 9, 17, 18].
III- THE ALGORITHM
We first present the "analytical" algorithm, i.e., the form of the algorithm
which involves no spatial discretization. Let aT : (0, oo) -• (0, oo) be a function
such that lim aT = 0. Let t >0 be fixed, and let the time step x= t/n where n^ 1is
T-»0
an integer. We consider the following algorithm;
(u*)= 0, u°=u0, (8)
that is nk+1
is determined from uk
by
uk + 1
= F(T)u (9a)
where
= q>+— [S(aT)/(<p)-/(<p)] for cpeL1
^). (9b)
We define the approximation un(t) to u(t) [the generalized solution of (4)] by
Concerning the convergence of un(t) to the solution u(t) of (4), one has.
THEOREM 1 : Assume u0 e LOT
(Q), set M = || u01| «j, anrf let i dénote the Lipschitz
constant of f on the interval [— M, M]. Assume the following stability condition
holds;
T ^ 1 for each x>0 (11)
{for example, this is valid if<jx = ix). Then lim un(t) = u{t) [the solution of (A)] in
R.A.LR.O. Analyse numérique/NumericaÏ Analysis
6. METHOD FOR SOLVING THEPROBLEM Ut ~ A / ( u )= 0 301
L1
(Q); inadditiontheconvergence isuniformfor t inanygivenbounded interval.
The proof will be obtained by demonstrating that F obeys a maximum
principle (lemma 1), that F is contractive in L1
(Q) (lemma 2), andby then
applying the nonlinear Chernoff formula.
LEMMA 1: If (11)isvalid,then -M^uk
<.Mfor all k.
Proof: We argue by induction; assume — M^uk
^M. Since the function
r -• r— x ƒ(r)/ax isnondecreasing [by (11)], it follows that
-M-if(-M)/<jx^uk
-Tf(uk
)/G^M-Tf{M)/ox. (12)
On theother hand, since ƒ is nondecreasing one has f( — M)?^f{uk
)Sf(M).
It follows from (2) that
f(-M)^S(oz)f(uk
)Sf(M). (13)
Combining (12)and (13), one obtains - M^ uk+x
S M. Therefore in performing
the itération (9), wecan replace ƒ byƒ where ƒ—ffor — M^r^M, f=f (M)for
r^.M, andƒ =ƒ (— M)for rg — M.In what follows wemaythus assume that ƒis
Lipschitz continuous with Lipschitz constant ionallof R1
.
LEMMA 2://(11) is valid, then F(x) is a contraction on L1
(Q), i.e.t
^^^x for
Proof: Indeed, we have
^ . (14)
Since the functions ƒ(r) and r—
x f(r)lox are nondecreasing in r,itfollows that
for r, s in H1
.
^ | / ( r ) - / ( s ) | + |(r-s)-^-(/(r)-/(5))| = | r - s | . (15)
Combining (14) and (15) gives the result.
Weconclude theproof by applying theorem 3.2 ofBrezis and Pazy [10](thisis
the nonlinear Chernoff formula). It suffices toverify that forevery cpeL1
and
every X>0:
£ ) x
^ (16)
vol. 13, n° 4,1979
7. 302 A. E. BERGER, H. BRÉZIS, J. C. W. ROGERS
in L1
(Q) as x -> 0. We have
x|/T+^(/-F(x))il/t = cp. (17)
x
-(/-F(x))i|r = q>T
>
then <pT ~* |/H-À,L/(|/) —|/ + A,>l|/~q> in Z,*(Q) as x -•0 since tyeD(A) implies
/(|/)e/)(L)and then
!
~ S
^ in LL
(Q) as x->0.
Finally, recalling that F(x) is contractive, combining (17) and (18) gives
Hence |||/T — ^||i = || <
p — <pT||i -*0 as x ^ O , and the proof of theorem 1 is
complete.
REMARKS: The conclusion of theorem 1 holds (with the same proof) if in the
scheme (9) one redefines F (x)<
p to be
(19)
where J (o) = (I + o L)"1
. This is an "analytical" version of the Laplace
modified forward Galerkin method considered by Douglas and Dupont in [16].
A similar approach may be used [3] to establish convergence of the truncation
method ([5, 6, 8]) for obstacle variational inequalities.
IV. NUMERICAL IMPLEMENTATION
Note that while the proofs in the previous section only treat the case of zero
boundary data, the extension of the algorithm given below for the situation with
nonzero Dirichlet data seems intuitively reasonable. We will discuss
implementation of the algorithm for the foliowing problem (for simplicity taking
ut = Af{u) in Q (20a)
u(p, £ = 0)= uo(p) in Q (20b)
"(P. t) = g(p, t) for peT, t>0. (20c)
R.A.LR.O. Analyse numérique/Numerical Analysis
8. METHOD FOR SOLV1NG THE PROBLEM M t ~ A / ( u ) = 0 303
Suppose one has approximate solution values {U"}at time tn
= nAt on a set of J
grid points { p,-}c Q(e.g. [/? = u0 (pj), j=l, ..., J). To obtain the approximate
solution values {l/"+ 1
} at the next time Ievel f"+1
= tn
-hA£, one performs the
following "discretization of (9)" (hère o
ccorresponds to aT/x):
/(t/7)
set 6
" ~ V ^ 7 = 1. ...,Jf (21a)
solve, using any appropriate numerical method, the linearheat équation (21a)
Ôr= aAg in Q, (22a)
n
)= Ô? for 7=1 ' . (22c)
obtaining values {ô"+
*}a t
the underlying grid points {pj} at t = tn
+ Ar; then
fff/")
I/j+1
= l/»+ Qj+1
-'/
-ï_ii for ; such that p;GQ, (21c)
and
E/ï+ 1
=ff(Pj, t"+1
) w h e n
P j e r
- (21d)
REMARKS: Assume there is a constant M such that g(p, t)g M, uo(p)^M,
and let |i be a Lipschitz constant for f(r) on —M^r^M. As before, by
modifying ƒ (r) on | r | > M, wemay suppose n is a Lipschitz constant for ƒ on all
of R1
. Then analogous to lemmas 1and 2; if a^ji and if the numerical method
used to solve (22) is Ll
stable, then so is the entire algorithm (21). That is, if
[ƒ"={(/"} and Un
are two different starting values, then
If a^n and the numerical scheme used to solve (22) satisfies a maximum
principle, then so does the entire algorithm (21), i.e.,
min(u0, g)SUn+i
^max{uOt g).
For gênerai L the appropriate maximum principle is | Un+1
a0^M. Sufficient
conditions for l1
stability and for a maximum principle for some standard finite
différence schemes and for piecewise linear finite éléments for (22) are given in,
e.g., [7].Wenote that Z1
stability or a maximum principle for the method used to
solve (22) is not necessary for reasonable numerical behavior of the
algorithm (21); similarly it is not always necessary to have oc^ia. (cf. the
vol. 13, n° 4, 1979
9. 304 A. E. BERGER, H. BRÉZIS, J. C. W. ROGERS
numerical examples below). Indeed, in developing error estimâtes for the
Laplace modified forward Galerkin équation when the solution u is "smooth",
Douglas and Dupont only require a>|i/2 [b].
V. NUMERICAL EXPERIMENTS
The first numerical experiments discussed will be for a problem with
/(M) = W|M| whose solution (on 0^x^20) is depicted by the solid Unes in
figure 1. The exact solution u (and thus u0 and g)was obtained from the top of
Figure 1. - Exact solution (solid lines) and numerical solution values (points) at t-0, t= .559
(diamonds), t = 2.23 (solid circles), t = H94 (open circles), and £=143. (triangles). The
approximate solution values were obtained using the algorithm (21) with a = l . , and using
the standard Crank-Nicolson finite différence method with 41 grid points and At = 143. /1024 to
solve (22).
page 363 of [17] (note that the m appearing as an exponent of the term
(m-l)/(2m(m+l)) in the définition of /(-q) should be omitted). Here x = 6
corresponds to x = 0 in [17], the solution reflects across x—6 via
u(6— r)— — u(6-hr),and the values for the parameters of [17]are m= 2,x= 1,and
a = 3.5 (the Lipschitz constant xfor this problem is a little less than 1.0).
The first numerical results for this problem (table I) that we will discuss were
obtained by solving (22) using the standard Crank-Nicolson (C.N.) finite
différence method using J = 433(equally spaced) grid points on [0, 20],and using
R.A.I.R.O. Analyse numérique/Numerical Analysis
10. METHOD FOR SOLVING THE PROBLEM Ut — Af(u) = 305
a variable number N of time steps to reach time T= 8.9375 (thus At= T/N). The
discrete L1
error Ex at time T was calculated by
£i =
20
Suppose J is large enough so that the spatial discretization error is relatively
negligible, and assume that Et is governed by
E^C^AtY, (23)
where Cx is a constant independent of At. Then the numerical rate of
convergence p computed from the results of using two successive values of N
(i.e. N and N = 2N with corresponding errors Ex and Ex) is
Results of taking ot=l, a= .75, and oc= .5 are given in table I (setting oc= .4
resulted in a fatal instability (exponent overflow) at N = 256). Note the decrease
in Ex (i.e. decrease in Cx of (23) while p remains roughly constant) as a is
decreased (until instability develops). The last result in table I was obtained by
TABLE I
Discrete Ll
errors and numerical convergence rates
for severai implementations of the algorithm (21)
N
4
8. . .
16
32
64. .
128
256. .
a=l.C.N.
Error
.2749
.1580
.0875
.0475
.0256
.0136
.0071
Rate
P
.80
.85
.88
.90
.91
.93
a=.75 C.N.
Error
.1666
.1026
.0583
.0321
.0174
.0093
.0049
Rate
P
.70
.82
.86
.89
.91
.93
oc=.5C
Error
.0996
.0541
.0306
.0171
.0093
.0050
.0026
:
.
N
.
Rate
P
.88
.82
.84
.87
.91
.92
a — . 5 Implicit
Error
.2747
.1578
.0873
.0474
.0255
.0136
.0071
Rate
P
.80
.85
.88
.90
.91
.93
using the standard fully implicit finite différence scheme (instead of C.N.) with
J = 433 to solve (22), and by taking a = . 5 (this corresponds to an
vol. 13, n° 4, 1979
11. 306 A. E. BERGER, H. BRÉZIS, J. C. W. ROGERS
implementation of a form of the Laplace modified forward Galerkin method
— c.f. p. 154of [16]).Notice the almost exact correspondence between the results
for (a = .5, implicit) and those for (a— 1, C.N.). This correspondence also occurs
between (a = .375, implicit) and (a= .75, C.N.), and between (oc = .25, implicit)
and (a = .5, C.N.) (J =433 throughout). Taking a = .2 with the implicit method
(J = 433) yields instability at A
f= 256. The plausibility of this correspondence
may be seen by writing out the algorithm (21) using the appropriate matrix K
for the scheme being used for (22), and here taking Un
to be the vector of values
{ L/"; pjGT}, etc. and assuming zero Dirichlet data
ƒ(£/") -f{U»)
For C.N., 9= 1/2, while for implicit, 0= 1 in this expression.
Alittle algebra shows that taking (C.N., a = a )is identical to taking (implicit,
a = a /2). Some more algebra shows that including the effect ofnonzero Dirichiet
data yields a formally O(At) différence between (C.N,, a = a) and (implicit,
<
x = a/2). Since the boundary data for this example is zero until t is near
r=8.9375, the numerical agreement is very pronounced, despite the fact that
the observed numerical error of the algorithm itself is behaving almost like
O (At). As an indication of the effect of the size of J on the error; with N = 256,
a= .5, and using C.N. for (22), the values of £, corresponding to several values
of J were;
f^.0177
109
.0044
217
.0042
433
.0026
For comparison, we also implemented the 3 Ievel (centered) Laplace modified
Galerkin équation {see p. 155 of [16]) for (20), in the foliowing form [here
Dl Uj^iUj-i-2 Uj+ *7j+1)/Ax2
and Ax is the (uniform) mesh length];
+ 2AD2
x(f(U'})) forysuchthatO<p;<20. (24)
The exact solution was used to provide the values U) at t= At. Results are given
in table II (again J = 433 and T=8.9375). The condition on the parameter p
given in [16] (for "smooth" problems) is p> |x/4 (here (3= .25 satisfies this while
R.A.I.R.O. Analyse numérique/Numerical Analysis
12. METHOD FOR SOLVING THE PROBLEM Ut - Af(u) = 307
P=.2 does not). Taking p = .15 yields instabiiity (exponent overflow) at
N =256. For N = 256 and P= .25 one has;
7 =55
Ei =.0163
109
.0028
217
.0023
433
.0005
Even for this (nonsmooth) problem, the 3 level scheme produces a higher
numerical rate of convergence.
TABLE II
Discrete L1
errors and numerical convergence rates
for several implementations of the Laplace modified centered équation (24)
N
4
8
16
32
64
128
256
P=-
Error
Ei
1.572
.4853
.1637
.0609
.0224
.0079
.0021
1
_
_
^ —
Rate
P
1.7
1.6
1.4
1.4
1.5
1.9
Error
Ei
.7560
.2160
.0854
.0349
.0139
.0048
.0009
5
Rate
P
1.8
1.3
1.3
1
.
3
1.5
2.4
p=.25
Error
.3175
.1179
.0485
.0212
.0084
.0024
.0005
Rate
P
1.4
1.3
1.2
1.3
1.8
2.3
Error
Ex
.2537
.1043
.0475
.0193
.0072
.0018
.0005
l
Rate
P
1.3
1.1
1.3
1.4
2.0
2.0
The second problem to be considered is a two phase Stefan problem whose
solution (in terms of température) is depicted in ïigtwe 2 (the exact solution is
given in both [7] and [20] —note that in this panicular example x= ki/Ci and
z= z"=ri = 0).In order to suppress the effect of thejump discontinuity from 0 toX
in enthalpy at the interface, errors discussed hère for the Stefan problem will be
errors in température values. The température v corresponding to an enthalpy
value u is easily obtained by inverting (6).
For the one phase Stefan problem, the alternating phase truncation (APT)
method ([7, 20]) reduces to (21) with a = i. In [20], for the "analytical" APT
method for a one dimensional one phase Stefan problem, it was shown that the
vol. 13. n° 4, 1979
13. 308 A. E. BERGER, H. BRÉZIS, J. C. W. ROGERS
Figure 2. —Exact solution (température) of a two phase Stefan problem (solid Unes) and numerical
solution values (points) at t = 0, t = 773 (triangles), t = 2 773 (open circles), and t = T (solid circles)
where T= 200,000. The approximate solution values were obtained using the algorithm (21) with
a = kjclt and using the standard Crank-Nicolson finite différence method with 41 grid points and
At= 772,595 tosolVe (22).
error in the température value and in the interface location could be bounded by
Un(l + T/At))1/2
, (25)
where T is the time at which one is bounding the error, and C is a constant
independent of At (but depending on T and the data of the problem). Let N
dénote the integer T/At, so then f*=N At=T. We have tested to see if the
temporal error behavior for (21) applied to the two phase Stefan problem is
consistent with (25) by fixing T, taking J "large", and calculating numerical
values for C for several values of Àt. Let v(x, t*) dénote the exact température
solution at time t? = Tt and let {Vf} dénote the approximate température
solution values at the grid points {pj} at time tN
obtained via (21) [i.e. V1
-is
defined to be the température corresponding (by (6)) to Uf]. The discrete Ll
error ex is defined to be
e =.
20 i.
and the numerical value of C corresponding to N (i.e. to At = T/N) is
(26)
(27)
R.A.I.R.O. Analyse numérique/Numerical Analysis
14. METHOD FOR SOLVÏNG THE PROBLEM Ut — A ƒ (tt) = 0 309
Results of applying the algorithm (21) to the two phase Stefan problem are
given in tables III and IV In all the tables for the Stefan problem
T= (2/3) 200,000 For table III, the standard Crank-Nicolson (C N ) method
TABLE III
Discrete L1
errors e1 and values of the constant C of (21)for several implementations of the algorithm
(21) and an implementatwn of the APT method for the two phase Stefan problem
A
216
432
864
1728
3 456
a = l 3n
ei j
9 10
6 81
4 75
3 46
2 43
C N
C
16
16
15
14
14
a = n C
10 16
6 63
4 40
3 06
2 22
N
C
18
15
14
13
13
a= 96 n
«i
84 6
89 8
88 9
88 2
82 1
C N
C
1 5
2 1
2 8
3 7
4 6
APTC
12 26
8 69
5 87
4 20
2 99
N
C
21
20
18
18
17
TABLE IV
Discrete L1
errors et and values of the constant C of {21)for several implementations of the algorithm
(21) for the two phase Stefan problem The Standard implicit method with J = 161 was used to
solve (22)
N
ot= 48 n
216
432
864
1728
3 456
1125
8 05
5 82
4 19
3 00
20
19
18
17
17
12
83
75
46
43
16
16
15
14
14
1025
6 76
4 39
3 02
2 22
18
16
14
13
13
92 0
92 3
89 7
87 4
82 1
1 6
2 1
2 8
3 6
4 6
with J = 161wasused to solve (22),results are given for a = 1 3,1,and 95 u, and
for companson, results for the APT method (C N ,J = 161) are given m the last
column The error behavior appears consistent with (25) The error decreases as
o
cdecreases until the abrupt détérioration below o
c = u Results when usmg the
Standard implicit method for (22) with J = 16i are given m table IV The
correspondence between (CN,a) and (implicit, a/2) is again observed [this is
not surpnsmg since O (At) is "small" relative to (25)] The data m table V
mdicates that for the values of JV being considered, the spatial discretization
error is relatively neghgible even for J = 41 We note that the method for the
Stefan problem discussed m [2] and [21] corresponds to using a purely explicit
method to solve (20) The 3 level Laplace modified Galerkin method (24) when
vol 13 n° 4 1979
15. 310 A. E. BERGER, H. BRÉZIS, J. C, W. ROGERS
applied to the Stefan problem in the form (20)produced very erratic behavior
with respect to variations in both P and J (typical behavior is presented in
table VI). Fatal instability occurs with J=161, p= .3(a,N= 3,456.
As an illustration of the simplicity of using (21)in several space dimensions,
numerical results for a twodimensional Stefan problem aregiven infigure 3. The
LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLILL
LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLt
LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLL
LLLLLL-LLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLL.LLLLLLL
LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLL
LLLLLi-LLi-LLLLlLLLLlLJÊ
LLLLLLLLLLLLLLLLLLL/S
LLLLLLLL LLLLLLL LL_LJSS
LLLLLLLLLLLLLLLLLL£SS
LLLLLLLLLLLLLLLLLl/SSS
L L L L L L L L L L ' - U L L L L L L L L L
LLLLLLLLLLLLLLLLLLuLL
LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLL LLLLLLLLLLc
LLLLLLLLLLLLLLLLLLLLL
L L L L L ' - L L - L L L L L L L L ^ L L L
LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLÉ
LLLLLLLLLLLLLLLLLtL/S
LLLLLLLL LLLLLL LLLLL/S
LLLLLL LLLLLLLLLLLL h S
/
LLLLLLLLLLLLLLLLL/S3S
LL LLLLLL L L L L L L L L / 3 SCi
LLLLLLLLL L L /
LLLLLLLLLLLLLLL/
LLLLLLLLLLLLLL/SSbSSS
/
LLLLLL LLLLLLI-L L L L L L L L
L L L L ' - L L L L L L L L L L L L L L I L
LLLLLLLLLLLL LLLLLL LL f
LLLLLLLL LLLL LL LL LL Lij
LLLLLLLLLLLL-LLLLLL
LuLLLLLLL'.LL LLLLLL J
LLLLLLLL LLLL-.LLLLL/
LLLLLL'.L LLLLLL LLL/SSÎ,
LLLLLLLLLLLLLLLL1/
LLLLLLLL LLLLLLLL/S S--s
L L L L L L ' - L L L L L L L L ' J SS31
LLLLLLLLLLLLLLL/ SbSSJ
LLLLLLLLLLLLL
LLLLuLLLLLLI
LLLLLLLLLLLI
LLLLLLLLLLLI
LLLLLLLLLLLI
t = T/3
LLLLLLLLLLL
L L L L L L L L L L L L L L L L L / S S S
LLLLLLLLLLLLLLLL/5SSJ
LLLLLLLLLLLLLLLU
LLLLLLLLLLLLLLL/ S3S*ÎS
LLLLLLLLLLLLLLt/ S^SSS
LL LL LL LL LL LL LL/ 5S - S
S S
LLLLLLLLLLLLL/SSSSSSS
L L L L L L L L L L L L L / S^S5r
S5
LLLLLLLLLlLL/"SSS
LLLLLLLLLLLL
LLLLLLLLLLy
LLLLLL Lt LL/S3^S"SSSS 5
LL L L L L LL L/IC "S <ÎS SS SSS
LLLLLLLLL/ESSSS-^'ÎSSS
LLtLLLLL/S^SSS^'SSSSS
LLLLLLLL
LLLLLLL S SSS3 S3 SS5
LLLLL LLLLLLLL LLLLI/S^S
LLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLL/
LLLLLLLLLLLLLLL i
LLLLLLLLLLLLLLL/
LLLLLLLLLLLLLL,
LLLLLLLLLLLLLL/
LLLLLLLLLLLLL/
LLLLLLULLLL
LLLLLLLLLLLL^
LLLLLLLLLLL
LLLLLLLLLLL/
LLLLLLLLLLy
LLLLLLLLLL/
LLLLLLLLL>
LLLLLLLL
LLLLLLLL
LLLLLLL
LLLLLLL
LLLLLL
LLLLLL
t=T
Figure 3. - Numerical solution of a two phase Stefan problem on the région O^x, y^20 using the
algorithm (21)with a = /c,7c,-. In each frame, an L(S)wasprinted at points where the approximate
solution value was ^ A.(^ 0). A blank wasprinted forvalues between 0 and X.Thesolid line isthe
exact interface location. Thestandard ADI method [19]with Ax= Ay = 1, wasused to solve(22).
For the first four frames the value of At used was 775,184 (7=200,000). For the frame onthe
lower rigbt the value of At was 77648.
classical ADImethod [19] wasused tosolve (22).Theexact solution at (x, y, t) is
given by u(p= x cos(30°) - y. sin(30°), t)where u is thesolution in [7],[20](p is
the projection of(x, y) on the line 30°below the positive x-axis, note also that
i=ki/ci and z= z—r1 =0).
R.A.Ï.R.O. Analyse mimérique/Numerical Analysis
16. METHOD FOR SOLVING THE PROBLEM Ut - Af(u) = 311
TABLE V
Discrete Ll
errors ex andvalues of the constant C of(21)for several implementations of the algorithm
(21) with a - ifor the twophase Stefan problem. TheCrank-Nicolson method was used to solve (22),
with several different values for J.
N
216
432
864
1,728
3,456.
J = 41
ei
8.64
6.44
4.12
3.67
2.51
C
.15
.15
.13
.15
.14
J-81
ei
9.33
6.15
4.28
3.12
2.34
L
c
.16
.14
.13
.13
.13
J = 16
ei
10.16
6.63
4.40
3.06
2.22
1
C
.18
.15
.14
.13
.13
TABLE VI
Discrete L1
errorsfor several implementations of the Laplace modified centered équation (24) for the
two phase Stefan problem in the farm (20).
JV
216
432
864
1,728
3,456
ei
48.88
27.34
10.35
1.48
.88
J=161
ei
57.30
27.81
5.47
1.29
.51
ei
39.22
23.05
3.69
1.59
.44
ei
68.56
28.69
4.16
.87
.27
J =81
ei
34.31
9.78
2.05
1.40
.65
38.41
6.16
1.38
.92
.60
ei
52.63
41.25
18.39
3.63
1.02
REFERENCES
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,
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August 12-16, 1974, L. CESARI, J. K. HALE and J. P. LASALLE, Eds, New York,
Academie Press, 1976, pp. 131-165.
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Academie Press, 1971, pp. 157-179.
14. M. G. CRANDALL and T. M. LIGGETT, Générationof Semi-groups of Nonlinear
Transformations on General Banach Spaces, Amer. J. Math., Vol. 93, 1971,
pp. 265-298.
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Media Equation, J. Math. Anal. Appl., Vol. 55, 1976, pp. 351-364.
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Media Equation II, J. Math. Anal. Appl., Vol. 57, 1977, pp. 522-538.
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Elliptic Differential Equations, J. Soc. Indust; Appl. Math., Vol. 3, 1955, pp. 28-41.
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R.A.I.R.O. Analyse numérique/Numerical Analysis