Finite Difference Discretization
of Hyperbolic Equations:
Linear Problems
Lectures 8, 9 and 10
First Order Wave Equation
INITION BOUNDARY VALUE PROBLEM (IBVP)
Initial Condition:
Boundary Conditions:
First Order Wave
Equation
Solution
Characteristics
General solution
First Order Wave
Equation
Solution
First Order Wave
Equation
Solution
First Order Wave
Equation
Stability
Model Problem
Initial condition:
Periodic Boundary conditions:
constant
Model Problem
Example
Periodic Solution (U>0)
Finite Difference
Solution
Discretization
Discretize (0,1) into J equal intervals
And (0,T) into N equal intervals
Finite Difference
Solution
Discretization
Finite Difference
Solution
Discretization
NOTATION:
•
•
•
approximation to
vector of approximate values at time ;
vector of exact values at time
;
Finite Difference
Solution
Approximation
For example … for ( U > 0 )
Forward in Time Backward (Upwind) in Space
Finite Difference
Solution
First Order Upwind Scheme
suggests …
Courant number C =
Finite Difference
Solution
First Order Upwind Scheme
Interpretation
Use Linear
Interpolation
j – 1, j
Finite Difference
Solution
First Order Upwind Scheme
Explicit Solution
no matrix inversion
exists and is unique
Finite Difference
Solution
We can write
First Order Upwind Scheme
Matrix Form
Finite Difference
Solution
First Order Upwind Scheme
Example
Convergence
Definition
The finite difference algorithm converges if
For any initial condition
.
Definition
Consistency
The difference scheme
,
is consistent with the differential equation
if:
For all smooth functions
when
.
First Order Upwind Scheme
Consistency
Difference operator
Differential operator
First Order Upwind Scheme
Consistency
First order accurate in space and time
Truncation Error
Insert exact solution
Consistency
into difference scheme
Definition
Stability
The difference scheme
There exists
for all
is stable if:
such that
; and n,
such that
Above condition can be written as
First Order Upwind Scheme
Stability
First Order Upwind Scheme
Stability
Stability
Stable if
Upwind scheme is stable provided
Lax Equivalence
Theorem
A consistent finite difference scheme for a partial
differential equation for which the initial value
problem is well-posed is convergent if and only if
it is stable.
Lax Equivalence
Theorem
Proof
( first order in
,
)
Lax Equivalence
Theorem
• Consistency:
• Stability:
•
Convergence
First Order Upwind Scheme
for
or
and
are constants independent of
,
Lax Equivalence
Theorem
Solutions for:
(left)
(right)
Convergence is slow !!
First Order Upwind Scheme
Example
Domains of dependence
CFL Condition
Mathematical Domain of Dependence of
Set of points in
where the initial or boundary
data may have some effect on
.
Numerical Domain of Dependence of
Set of points in
where the initial or boundary
data may have some effect on
.
Domains of dependence
CFL Condition
First Order Upwind Scheme
Analytical
Numerical ( U > 0 )
CFL Condition
CFL Theorem
CFL Condition
For each
the mathematical domain of de-
pendence is contained in the numerical domain of
dependence.
CFL Theorem
The CFL condition is a necessary condition for the
convergence of a numerical approximation of a partial
differential equation, linear or nonlinear.
CFL Condition
Stable
CFL Theorem
Unstable
Fourier Analysis
• Provides a systematic method for determining
stability → von Neumann Stability Analysis
• Provides insight into discretization errors
Fourier Analysis
Continuous Problem
Fourier Modes and Properties…
Fourier mode:
• Periodic ( period = 1 )
• Orthogonality
•Eigenfunction of
( integer )
Fourier Analysis
Continuous Problem
…Fourier Modes and Properties
• Form a basis for periodic functions in
• Parseval’s theorem
Fourier Analysis
Continuous Problem
Wave Equation
Fourier Analysis
Discrete Problem
Fourier Modes and Properties…
Fourier mode:
k ( integer )
,
Fourier Analysis
Real part of first 4
Fourier modes
Discrete Problem
…Fourier Modes and Properties…
Fourier Analysis
• Periodic (period = J)
• Orthogonality
Discrete Problem
…Fourier Modes and Properties…
Fourier Analysis
Discrete Problem
…Fourier Modes and Properties…
• Eigenfunctions of difference operators e.g.,
Fourier Analysis
Discrete Problem
Fourier Modes and Properties…
• Basis for periodic (discrete) functions
• Parseval’s theorem
Fourier Analysis
Write
Stability
Stability for all data
von Neumann Stability Criterion
Fourier Analysis
von Neumann Stability Criterion
First Order Upwind Scheme…
Fourier Analysis
von Neumann Stability Criterion
…First Order Upwind Scheme…
amplification factor
Stability if
which implies
Fourier Analysis
Stability if:
von Neumann Stability Criterion
…First Order Upwind Scheme
Fourier Analysis
von Neumann Stability Criterion
FTCS Scheme…
Fourier Decomposition:
Fourier Analysis
von Neumann Stability Criterion
…FTCS Scheme
amplification factor
Unconditionally Unstable
Not Convergent
Lax-Wendroff
Scheme
Time Discretization
Write a Taylor series expansion in time about
But …
Lax-Wendroff
Scheme
Spatial Approximation
Approximate spatial derivatives
Lax-Wendroff
Scheme
Equation
no matrix inversion
exists and is unique
Lax-Wendroff
Scheme
Interpretation
Use Quadratic
Interpolation
Lax-Wendroff
Scheme
Analysis
Consistency
Second order accurate in space and time
Lax-Wendroff
Scheme
Analysis
Truncation Error
Insert exact solution into difference scheme
Consistency
Lax-Wendroff
Scheme
Stability if:
Analysis
Stability
Lax-Wendroff
Scheme
Analysis
Convergence
• Consistency:
• Stability:
•
Convergence
and
are constants independent of
Lax-Wendroff
Scheme
Analytical
Domains of Dependence
Numerical
Lax-Wendroff
Scheme
Stable
CFL Condition
Unstable
Lax-Wendroff
Scheme
Solutions for:
C = 0.5
= 1/50 (left)
= 1/100 (right)
Example
Lax-Wendroff
Scheme
= 1/100
C = 0.5
Upwind (left)
vs.
Lax-Wendroff (right)
Example
Beam-Warming
Scheme
Derivation
Use Quadratic
Interpolation
Beam-Warming
Scheme
• Consistency,
• Stability
Consistency and Stability
Method of Lines
Generally applicable to time evolution PDE’s
• Spatial discretization
Semi-discrete scheme (system of coupled
ODE’s
• Time discretization (using ODE techniques)
Discrete Scheme
By studying semi-discrete scheme we can better
understand spatial and temporal discretization errors
Method of Lines
Notation
approximation to
vector of semi-discrete approximations;
Method of Lines
Spatial Discretization
Central difference … (for example)
or, in vector form,
Method of Lines
Spatial Discretization
Fourier Analysis …
Write semi-discrete approximation as
inserting into semi-discrete equation
Method of Lines
Spatial Discretization
… Fourier Analysis …
For each θ, we have a scalar ODE
Neutrally stable
Method of Lines
Exact solution
Semi-discrete solution
Spatial Discretization
… Fourier Analysis …
Method of Lines
Spatial Discretization
Fourier Analysis …
Method of Lines
Time Discretization
Predictor/Corrector Algorithm …
Model ODE
Predictor
Corrector
Combining the two steps you have
Method of Lines
Time Discretization
…Predictor/Corrector Algorithm
Semi-discrete equation
Predictor
Corrector
Combining the two steps you have
Method of Lines
Fourier Stability Analysis
Fourier Transform
Method of Lines
Application factor
with
Stability
Fourier Stability Analysis
Method of Lines
Fourier Stability Analysis
PDE
Semi-discrete
Discrete
Semi-discrete Fourier
Discrete Fourier
Method of Lines
Semi-discrete
Fourier semi-discrete
Predictor
Corrector
Discrete
Fourier Stability Analysis
Path B …
Method of Lines
Fourier Stability Analysis
…Path B
• Give the same discrete Fourier equation
• Simpler
• “Decouples” spatial and temporal discretization
For each θ, the discrete Fourier equation is the
result of discretizing the scalar semi-discrete
ODE for the θ Fourier mode
Method of Lines
Model equation:
Discretization
Methods for ODE’s
complex- valued
EF
EB
CN
Method of Lines
Given
Methods for ODE’s
Absolute Stability Diagrams …
and
complex-valued
(EF) or
is defined such that
(EB) or … ;
Method of Lines
Methods for ODE’s
…Absolute Stability Diagrams …
EF
EB
CN
Method of Lines
Methods for ODE’s
…Absolute Stability Diagrams
Method of Lines
Methods for ODE’s
Application to the wave equation…
For each
Thus,
•
•
(and
) is purely imaginary
for
Method of Lines
Methods for ODE’s
…Application to the wave equation…
EF is unconditionally unstable
EB is unconditionally stable
CN is unconditionally stable
Method of Lines
Methods for ODE’s
…Application to the wave equation…
Stable schemes can be obtained by:
1) Selecting explicit time stepping algorithm which
have some stability on imaginary axis
2) Modifying the original equation by adding “artificial
viscosity”
Method of Lines
Methods for ODE’s
…Application to the wave equation…
Explicit Time Stepping Scheme
Predictor/Corrector
Method of Lines
Methods for ODE’s
…Application to the wave equation…
Explicit Time Stepping Scheme
4 Stage Runge-Kutta
Method of Lines
Methods for ODE’s
…Application to the wave equation…
Adding Artificial Viscosity
Additional Term
EF Time
EF Time
First Order Upwind
Lax-Wendroff
Method of Lines
Methods for ODE’s
…Application to the wave equation…
Adding Artificial Viscosity
For each Fourier mode θ,
Additional Term
Method of Lines
Methods for ODE’s
…Application to the wave equation…
First Order Upwind Scheme
Method of Lines
Methods for ODE’s
…Application to the wave equation
Lax-Wendroff Scheme
Dissipation and
Dispersion
with
Solution
Model Problem
and periodic boundary conditions.
Dissipation and
Dispersion
Model Problem
represents Decay
dissipation relation
represents Propagation
dispersion relation
For exact solution of
no dissipation
(constant) no dispersion
Dissipation and
Dispersion
First Order Upwind
Lax-Wendroff
Beam-Warming
Modified Equation
Dissipation and
Dispersion
•
•
•
•
Modified Equation
For the upwind scheme dissipation dominates over
dispersion Smooth solutions
For Lax-Wendroff and Beam-Warming dispersion is
the leading error effect
Oscillatory solutions ( if
not well resolved)
Lax-Wendroff has a negative phase error
Beaming-Warming has (for
) a positive phase
error
Dissipation and
Dispersion
First Order Upwind
Examples
Dissipation and
Dispersion
Lax-Wendroff (left)
vs.
Beam-Warming (right)
Examples
Dissipation and
Dispersion
Exact Discrete Relations
For the exact solution
, and
For the discrete solution
, and