Lectures 8, 9 and 10

Lectures 8, 9 and 10 Finite Difference Discretization of Hyperbolic Equations:Linear Problems

First Order Wave Equation INITION BOUNDARY VALUE PROBLEM (IBVP) Initial Condition: Boundary Conditions:

Solution First Order Wave Equation Characteristics General solution

Solution First Order Wave Equation

Stability First Order Wave Equation

Model Problem Initial condition: Periodic Boundary conditions: constant

Example Model Problem Periodic Solution (U>0)

Discretization Finite DifferenceSolution Discretize (0,1) into J equal intervals And (0,T) into N equal intervals

Discretization Finite DifferenceSolution

Discretization Finite DifferenceSolution NOTATION: • approximation to • vector of approximate values at time ; • vector of exact values at time ;

Approximation Finite DifferenceSolution For example … for ( U > 0 ) Forward in Time Backward (Upwind) in Space

First Order Upwind Scheme Finite DifferenceSolution suggests … Courant number C =

First Order Upwind Scheme Finite DifferenceSolution Interpretation Use Linear Interpolation j – 1, j

First Order Upwind Scheme Finite DifferenceSolution Explicit Solution no matrix inversion exists and is unique

First Order Upwind Scheme Finite DifferenceSolution Matrix Form We can write

First Order Upwind Scheme Finite DifferenceSolution Example

Definition Convergence 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 into difference scheme Consistency

Definition Stability The difference scheme is stable if: There exists such that for all ; and n, such that Above condition can be written as

First Order Upwind Scheme Stability

Stability Stable if Upwind scheme is stable provided

Lax EquivalenceTheorem 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.

Proof Lax EquivalenceTheorem ( first order in , )

First Order Upwind Scheme Lax EquivalenceTheorem • Consistency: • Stability: for • Convergence or and are constants independent of ,

First Order Upwind Scheme Lax EquivalenceTheorem Example Solutions for: (left) (right) Convergence is slow !!

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 Theorem CFL Condition CFL Condition For each the mathematical domain of dependence 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 Theorem CFL Condition Stable Unstable

Fourier Analysis • Provides a systematic method for determining stability → von Neumann Stability Analysis • Provides insight into discretization errors

Continuous Problem Fourier Analysis Fourier Modes and Properties… Fourier mode: ( integer ) • Periodic ( period = 1 ) • Orthogonality • Eigenfunction of

Continuous Problem Fourier Analysis …Fourier Modes and Properties • Form a basis for periodic functions in • Parseval’s theorem

Continuous Problem Fourier Analysis Wave Equation

Discrete Problem Fourier Analysis Fourier Modes and Properties… Fourier mode: , k ( integer )

Discrete Problem Fourier Analysis …Fourier Modes and Properties… Real part of first 4 Fourier modes

Discrete Problem Fourier Analysis …Fourier Modes and Properties… • Periodic (period = J) • Orthogonality

Discrete Problem Fourier Analysis …Fourier Modes and Properties… • Eigenfunctions of difference operators e.g.,

Discrete Problem Fourier Analysis Fourier Modes and Properties… • Basis for periodic (discrete) functions • Parseval’s theorem

von Neumann Stability Criterion Fourier Analysis Write Stability Stability for all data

von Neumann Stability Criterion Fourier Analysis First Order Upwind Scheme…

von Neumann Stability Criterion Fourier Analysis …First Order Upwind Scheme… amplification factor Stability if which implies

von Neumann Stability Criterion Fourier Analysis …First Order Upwind Scheme Stability if:

von Neumann Stability Criterion Fourier Analysis FTCS Scheme… Fourier Decomposition:

von Neumann Stability Criterion Fourier Analysis …FTCS Scheme amplification factor Unconditionally Unstable Not Convergent

Time Discretization Lax-WendroffScheme Write a Taylor series expansion in time about But …

Lectures 8, 9 and 10