lecture 3. fd methods domain of influence (elliptic, parabolic, hyperbolic) conservative formulation...
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Lecture 3
FD Methods
Domain of Influence (elliptic, parabolic, hyperbolic)
Conservative formulation
Complications:
- Mixed Derivatives- Higher Dimensions (2+)- Source Terms
FD: Domain of influence
Elliptic PDE Laplace equation
0,U,U2
2
2
2
y)(xy
y)(xx
Hy
Wx
0
0
FD: Domain of influence
Parabolic PDE Heat equation
(t,x)x
k(t,x)t
UU2
2
)()0(
),(
)0( 0
xg,xU
uAtU
ut,U
A
FD: Domain of influence
Hyperbolic PDE Wave equation
(t,x)x
C(t,x)t
UU2
22
2
2
)()0(
)()0(
)0( 0
xg,xUt
xf,xU
ut,U
Mixed Derivatives
Central
Derivative
Stencil:
yx
),j(i),j(i
yx
),j(i),j(i(x,y)
yx
4
11U11U
4
11U11UU
2
PDE FormulationBurgers Equation:
Straightforward discretization (upwind):
Conservative Form:
Standard discretization (upwind):
0UUU
(t,x)x
(t,x)(t,x)t
(i,j))-(i,j(i,j)Δx
Δt(i,j) - ,j)(i U1UUU1U
0U2
1U 2
(t,x)x
(t,x)t
22 U1U2
U1U (i,j)-)(i,jΔx
Δt(i,j) - ,j)(i
Burgers Equation
Lax-Wendroff Theorem
For hyperbolic systems of conservation laws, schemes written in conservation form guarantee that if the scheme converges numerically, then it converges to the analytic solution of the original system of equations.
Lax equivalence:
Stable solutions converge to analytic solutions
Implicit and Explicit Methods
Explicit FD method:
U(i+1) is defined by U(i) knowing the values at time n, one can obtain
the corresponding values at time n+1
Implicit FD method:
U(i+1) is defined by solution
of system of equations at every
time step
Implicit and Explicit Methods
Crank-Nicolson for the Heat Equation
Implicit and Explicit Methods
+ Unconditionally Stable
+ Second Order Accurate in Time
- Complete system should be solved
at each time step
Crank-Nicolson method:
stable for larger time steps
First vs Second Order Accuracy
Local truncation error vs. grid resolution in x
Source Terms
Nonlinear Equation with source term S(U)
e.g. HD in curvilinear coordinates
1. Unsplit method
2. Fractional step (splitting method)
)U()U(U SFx
(t,x)t
Source Terms: unsplit method
One sided forward method:
- Lax-Friedrichs (linear + nonlinear)
- Leapfrog (linear)
- Lax-Wendroff (linear+nonlinear)
- Beam-Warming (linear)
)U(
)U()1F(UU1U
(i,j)ΔtS
(i,j)-F)(i,jΔx
Δt(i,j) - ,j)(i
Source Terms: splitting method
Split inhomogeneous equation into two steps:
transport + sources
1) Solve PDE (transport)
2) Solve ODE (source)
0)U(U
Fx
(t,x)t
(i,j)(i,j) UU
)U(U S(t,x)t
,j)(i(i,j)(i,j) 1UUU
Multidimensional Problems
Nonlinear multidimensional PDE:
Dimensional splitting
x-sweep
y-sweep
z-sweep
source term: splitting method
)U()U()U()U(U SHz
Gy
Fx
(t,x)t
Dimension Splittingx-sweep (PDE):
y-sweep (PDE):
z-sweep (PDE):
source (ODE):
0)U(U
Fx
(t,x)t
0)U(U **
Gy
(t,x)t
0)U(U ****
Hz
(t,x)t
)U(U ****** S(t,x)t
(i,j)(i,j) *UU
(i,j)(i,j) *** UU
(i,j)(i,j) ***** UU
,j)(i(i,j) 1UU ***
Dimension Splitting
Upwind method (forward difference)
)U(
)U()1(U
)U()1(U
)U()1(UU1U
(i,j)ΔtS
(i,j)-H)(i,jHΔz
Δt-
(i,j)-G)(i,jGΔy
Δt-
(i,j)-F)(i,jFΔx
Δt(i,j) - ,j)(i
Dimension Splitting
+ Speed
+ Numerical Stability
- Accuracy
Method of Lines
Conservation Equation:
Only spatial discretization:
Solution of the ODE (i=1..N)- Analytic?- Runge-Kutta
0)U(U
Fx
(t,x)t
)()(U jfjt
))(U())1(U(1
)( jFjFx
jf
Method of Lines
Multidimensional problem:
Lines:
Spatial discretazion:
)U()U()U(U SGy
Fx
(t,x)t
)),(U(),(),(),(U jiSjigjifjit
)),(U()),1(U(1
),( jiFjiFx
jif
)),(U())1,(U(1
),( jiGjiGy
jig
Method of Lines
Analytic solution in time: Numerical error only due to spatial discretization;
+ For some problems analytic solutions exist;
+ Nonlinear equations solved using stable scheme
(some nonlinear problems can not be solved using implicit method)
- Computationally extensive on high resolution grids;
Linear schemes
“It is not possible for a linear scheme to be both higher that first order accurate and free of spurious oscillations.”
Godunov 1959
First order: numerical diffusion;
Second order: spurious oscillations;
end
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