4. About PDEs :
A second order PDE :
Linear Equations:A,B,C,D,E and F are constants or f(x,y)
Non-linear Equations:A,B,C,D,E and F contain φ or its derivatives
Quasilinear Equations: Important subclass of nonlinear equations.A,B,C,D,E and
F may be function of φ or its first derivatives
Homogeneous: G=0
Parabolic Equation: B*B - 4AC = 0
Elliptic Equation: B*B – 4AC < 0
Hyperbolic Equation: B*B – 4AC > 0
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5. About PDEs :
Some basic equations :
Navier Stoke Equation : Elliptical in space ; Parabolic in time
Laplace and Poisson Equations: Elliptic
Fluid flow problems have non – linear terms called advection
and convection terms in momentum and energy equations
respectively.
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6. Various Discretization Methods
Finite Difference Method (FDM)
Finite Element Method (FEM)
FiniteVolume Method (FVM)
Spectral Method
Lattice Gas Cellular Automata (LGCA)
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7. FDM - Overview
Boundary Conditions
No. of Boundary conditions required is order of highest
derivative appearing in each independent variable
Unsteady equations governed by a first derivative in time
require initial condition to carry out the time integration
Three types of spatial boundary conditions:
Dirchlet Condition
Neumann Condition
Mixed Boundary Condition
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8. FDM - Overview
Basic Procedure
Replace derivatives of governing equations with algebraic
difference quotients
Results in a system of algebraic equations solvable for
dependent variables at discrete grid points
Analytical solutions provide closed-form expressions –variation
of dependent variables in the domain
Numerical solutions (finite difference) - values at discrete
points in the domain
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9. FDM - Overview
Discrete Grid Points
Δx and Δy –spacing in positive
x and y direction
Δx & Δy not necessarily uniform.
In some cases, numerical
calculations performed on trans-
formed computational plane
having uniform spacing in transformed variables but
non uniform spacing in physical plane.
Grid points identified by indices i and j in positive x and y
direction respectively.
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10. Basics of Finite Difference Methods
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11. Taylor Series Expansion
for small Δx higher order terms can be neglected.
n-order accurate Truncation error
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12. Finite Difference Quotients and
Truncation Error
Forward Difference Truncation Error 𝒪(Δx)
Backward Difference Truncation Error 𝒪(Δx)
Central Difference Truncation Error 𝒪(Δx)2
Central Difference for Second Derivative
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13. Finite Difference Quotients and
Truncation Error
Basic concept: Replace each term of PDE with its finite
difference equivalent term
Partial difference equation:
Using Forward time Central Space (FTCS) method of
discretization:
n: conditions at time t
i: grid point in spatial dimension
Truncation Error (TE) = 𝒪(Δt, (Δx)2)
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15. Explicit Method
Explicit method uses the fact that we know the
dependent variable, u at all x at time t from initial
conditions
Since the equation contains only one unknown, (i.e. u
at time t+Δt), it can be obtained directly from known
values of u at t
The solution takes the form of a “marching” procedure in
steps of time
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16. Crank – Nicolson Implicit Method
The unknown value u at time level (n+1) is expressed
both in terms of known quantities at n and unknown
quantities at (n+1).
The spatial differences on RHS are expressed in terms of
averages between time level n and (n+1) :
The above equation cannot result in a solution of at
grid point i.
The eq. is written at all grid points resulting in a system of
algebraic equations which can be solved simultaneously
for u at all i at time level (n+1).
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17. Crank – Nicolson Implicit Method
The equation can be rearranged as
where r = αΔt/(Δx) 2
On application of eq. at all grid points from i=1 to i=k+1 , the
system of eqs. with boundary conditions u=A at x=0 and u=D
at x=L can be expressed in the form of Ax = C
A is the tridiagonal coefficient matrix and x is the solution
vector.The eq. can be solved using Thomas Algorithm
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18. Explicit ~ Implicit – A Comparison
Explicit Method
Easy to set up.
Constraint on mesh width, time-step.
Less computer time.
Implicit Method
Complicated to set up.
Larger computer time.
No constraint on time step.
Can be solved using Thomas Algorithm.
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20. Consistency
A finite difference representation of a PDE is said to be
consistent if:
For equations where truncation error is 𝒪(Δx) or 𝒪(Δt)
or higher orders,TE vanishes as the mesh is refined
However, for schemes where TE is 𝒪(Δt/Δx), the scheme
is not consistent unless mesh is refined in a manner such
that Δt/Δx→0
For the Dufort-Frankel differencing scheme (1953), if
Δt/Δx does not tend to zero, a parabolic PDE may end up
as a hyperbolic equation
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21. Convergence
A solution of the algebraic equations that approximate a
PDE is convergent if the approximate solution approaches
the exact solution of the PDE for each value of the
independent variable as the grid spacing tends to zero :
RHS is the solution of algebraic equation
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22. Errors & Stability Analysis
Errors :
A = Analytical solution of PDE
D = Exact solution of finite difference equation
N = Numerical solution from a real computer with finite
accuracy
Discretization Error = A –D = Truncation error + error
introduced due to the treatment of boundary condition
Round-off Error = ε= N –D
N = ε+ D
ε will be referred to as “error” henceforth
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23. Errors & Stability Analysis
Consider the 1-D unsteady state heat conduction equation and its FDE :
N must satisfy the finite difference equation :
Also, D being the exact solution also satisfies FDE :
Subtracting above 2 equations, we see that error ε also satisfies FDE :
If errors εi‟s shrink or remain same from step n to n+1, solution is stable.
Condition for stability is :
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25. Introduction
Fluid mechanics: More complex, governing PDE‟s form a
nonlinear system.
Burger‟s Equation: => Includes time
dependent, convective and diffusive term.
Here „u‟: velocity,„γ‟: coefficient of viscosity, & „ζ‟: any
property which can be transported or diffused.
Neglecting viscous term, remaining equation is a simple
analog of Euler‟s equation :
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26. Conservative Property
FDE possesses conservative property if it preserves integral
conservation relations of the continuum
Consider VorticityTransport Equation:
where is nabla,V is fluid velocity and ω is vorticity.
Integrating over a fixed region we get,
which can be written as :
i.e. rate of accumulation of ω in is equal to net advective flux rate plus
net diffusive flux rate of ω across Ao into
The concept of conservative property is to maintain this integral
relation in finite difference representation.
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27. Conservative Property
Consider inviscid Burger‟s equation :
Evaluating the integral over a region running from i=I1
to i=I2 :
Thus, the FDE analogous to inviscid part of the
integral has preserved the conservative property.
For non-conservative form of inviscid Burger‟s equation:
i.e. FDE analog has failed to preserve the conservative property
FDE Analog
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28. Transportive Property
FDE formulation of a flow is said to possess the transportive property if
the effect of perturbation is convected only in the direction of velocity
Consider model Burger‟s equation in conservative form and a perturbation
εm = δ in ζ for u>0, all other ε=0
Using FTCS, we find the transportive property to be violated
On the contrary when an upwind scheme is used,
=> Downstream Location (m+1)
=> Point m of disturbance
=> Upstream Location (m-1)
Upwind method maintains unidirectional flow of information.
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29. The Upwind Method
The inviscid Burger‟s equation in the following forms are
unconditionally unstable :
The equations can be made stable by using backward space
difference scheme if u > 0 and forward space difference
scheme if u < 0 :
Upwind method of discretization is necessary in convection
dominated flows.
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30. References
Knabner P., Angerman L. - “Numerical Methods for Elliptic
and Parabolic Partial Differential Equations”
"Numerical simulation in fluid dynamics: a practical
introduction" by Michael Griebel, Thomas Dornseifer, Tilman
Neunhoeffe
http://www10.informatik.uni-
erlangen.de/en/Teaching/Courses/WS2011/SiWiR/material/scri
pt.pdf
"Introduction to Partial Differential Equations (A
ComputationalApproach)" by Aslak Tveito, Ragnar Winther
(Publisher: Springer Berlin)
“Finite Difference Schemes and Partial Differential
Equations, Second Edition” , John C. Strikwerda
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