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**Finite Difference Discretization of Hyperbolic Equations: Linear Problems**

Lectures 8, 9 and 10

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**First Order Wave Equation**

INITION BOUNDARY VALUE PROBLEM (IBVP) Initial Condition: Boundary Conditions:

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**First Order Wave Equation**

Solution First Order Wave Equation Characteristics General solution

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**First Order Wave Equation**

Solution First Order Wave Equation

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**First Order Wave Equation**

Solution First Order Wave Equation

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**First Order Wave Equation**

Stability First Order Wave Equation

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**Model Problem Initial condition: Periodic Boundary conditions:**

constant

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**Periodic Solution (U>0)**

Example Model Problem Periodic Solution (U>0)

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**Finite Difference Solution**

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

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**Finite Difference Solution**

Discretization Finite Difference Solution

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**Finite Difference Solution**

Discretization Finite Difference Solution NOTATION: approximation to vector of approximate values at time ; vector of exact values at time ;

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**Finite Difference Solution**

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

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**Finite Difference Solution**

First Order Upwind Scheme Finite Difference Solution suggests … Courant number C =

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**Finite Difference Solution**

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

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**Finite Difference Solution**

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

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**Finite Difference Solution**

First Order Upwind Scheme Finite Difference Solution Matrix Form We can write

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**Finite Difference Solution**

First Order Upwind Scheme Finite Difference Solution Example

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**Convergence Definition The finite difference algorithm converges if**

For any initial condition

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**Consistency Definition The difference scheme ,**

is consistent with the differential equation if: For all smooth functions when

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**First Order Upwind Scheme**

Consistency Difference operator Differential operator

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**First Order Upwind Scheme**

Consistency First order accurate in space and time

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**Truncation Error Insert exact solution into difference scheme**

Consistency

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**Stability Definition The difference scheme is stable if:**

There exists such that for all ; and n, such that Above condition can be written as

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**First Order Upwind Scheme**

Stability

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**First Order Upwind Scheme**

Stability

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Stability Stable if Upwind scheme is stable provided

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**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.

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**Lax Equivalence Theorem**

Proof Lax Equivalence Theorem ( first order in , )

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**Lax Equivalence Theorem**

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

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**Lax Equivalence Theorem**

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

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**CFL Condition Mathematical Domain of Dependence of**

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 data may have some effect on

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**First Order Upwind Scheme**

Domains of dependence CFL Condition First Order Upwind Scheme Analytical Numerical ( U > 0 )

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**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.

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CFL Theorem CFL Condition Stable Unstable

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

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**Fourier Modes and Properties…**

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

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**…Fourier Modes and Properties**

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

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Continuous Problem Fourier Analysis Wave Equation

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**Fourier Modes and Properties…**

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

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**…Fourier Modes and Properties…**

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

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**…Fourier Modes and Properties…**

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

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**…Fourier Modes and Properties…**

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

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**Fourier Modes and Properties…**

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

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**von Neumann Stability Criterion**

Fourier Analysis Write Stability Stability for all data

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**Fourier Analysis von Neumann Stability Criterion**

First Order Upwind Scheme…

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**Fourier Analysis amplification factor Stability if which implies**

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

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**Fourier Analysis Stability if: von Neumann Stability Criterion**

…First Order Upwind Scheme Stability if:

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**von Neumann Stability Criterion**

Fourier Analysis FTCS Scheme… Fourier Decomposition:

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**von Neumann Stability Criterion**

Fourier Analysis …FTCS Scheme amplification factor Unconditionally Unstable Not Convergent

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**Lax-Wendroff Scheme Time Discretization**

Write a Taylor series expansion in time about But …

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**Spatial Approximation**

Lax-Wendroff Scheme Approximate spatial derivatives

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Equation Lax-Wendroff Scheme no matrix inversion exists and is unique

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Interpretation Lax-Wendroff Scheme Use Quadratic Interpolation

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**Lax-Wendroff Scheme Analysis Second order accurate in space and time**

Consistency Second order accurate in space and time

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**Lax-Wendroff Scheme Analysis**

Truncation Error Insert exact solution into difference scheme Consistency

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Analysis Lax-Wendroff Scheme Stability Stability if:

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**Lax-Wendroff Scheme Analysis Consistency: Stability: Convergence**

and are constants independent of

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Domains of Dependence Lax-Wendroff Scheme Analytical Numerical

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CFL Condition Lax-Wendroff Scheme Stable Unstable

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**Lax-Wendroff Scheme Example Solutions for: C = 0.5 = 1/50 (left)**

= 1/100 (right)

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**Lax-Wendroff Scheme Example = 1/100 C = 0.5 Upwind (left) vs.**

Lax-Wendroff (right)

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Derivation Beam-Warming Scheme Use Quadratic Interpolation

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**Consistency and Stability**

Beam-Warming Scheme Consistency, Stability

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**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

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**Method of Lines Notation approximation to**

vector of semi-discrete approximations;

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**Spatial Discretization**

Method of Lines Central difference … (for example) or, in vector form,

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**Spatial Discretization**

Method of Lines Fourier Analysis … Write semi-discrete approximation as inserting into semi-discrete equation

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**Spatial Discretization**

Method of Lines … Fourier Analysis … For each θ, we have a scalar ODE Neutrally stable

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**Spatial Discretization**

Method of Lines … Fourier Analysis … Exact solution Semi-discrete solution

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**Spatial Discretization**

Method of Lines Fourier Analysis …

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**Predictor/Corrector Algorithm …**

Time Discretization Method of Lines Predictor/Corrector Algorithm … Model ODE Predictor Corrector Combining the two steps you have

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**…Predictor/Corrector Algorithm**

Time Discretization Method of Lines …Predictor/Corrector Algorithm Semi-discrete equation Predictor Corrector Combining the two steps you have

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**Fourier Stability Analysis**

Method of Lines Fourier Transform

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**Fourier Stability Analysis**

Method of Lines Application factor with Stability

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**Method of Lines PDE Semi-discrete Discrete Semi-discrete Fourier**

Fourier Stability Analysis Method of Lines PDE Semi-discrete Discrete Semi-discrete Fourier Discrete Fourier

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**Fourier Stability Analysis**

Method of Lines Path B … Semi-discrete Fourier semi-discrete Predictor Corrector Discrete

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**Fourier Stability Analysis**

Method of Lines …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

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**Method of Lines Methods for ODE’s Model equation: complex- valued**

Discretization EF EB CN

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**Absolute Stability Diagrams …**

Methods for ODE’s Method of Lines Absolute Stability Diagrams … Given and complex-valued (EF) or (EB) or … ; is defined such that

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**…Absolute Stability Diagrams …**

Methods for ODE’s Method of Lines …Absolute Stability Diagrams … EF EB CN

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**…Absolute Stability Diagrams**

Methods for ODE’s Method of Lines …Absolute Stability Diagrams

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**Application to the wave equation…**

Methods for ODE’s Method of Lines Application to the wave equation… For each Thus, (and ) is purely imaginary for

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**…Application to the wave equation…**

Methods for ODE’s Method of Lines …Application to the wave equation… EF is unconditionally unstable EB is unconditionally stable CN is unconditionally stable

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**…Application to the wave equation…**

Methods for ODE’s Method of Lines …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”

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**Method of Lines Methods for ODE’s Explicit Time Stepping Scheme**

…Application to the wave equation… Explicit Time Stepping Scheme Predictor/Corrector

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**Method of Lines Methods for ODE’s Explicit Time Stepping Scheme**

…Application to the wave equation… Explicit Time Stepping Scheme 4 Stage Runge-Kutta

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**Method of Lines Methods for ODE’s Adding Artificial Viscosity**

…Application to the wave equation… Adding Artificial Viscosity Additional Term EF Time First Order Upwind EF Time Lax-Wendroff

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**Method of Lines Methods for ODE’s Adding Artificial Viscosity**

…Application to the wave equation… Adding Artificial Viscosity For each Fourier mode θ, Additional Term

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**…Application to the wave equation…**

Methods for ODE’s Method of Lines …Application to the wave equation… First Order Upwind Scheme

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**…Application to the wave equation**

Methods for ODE’s Method of Lines …Application to the wave equation Lax-Wendroff Scheme

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**Dissipation and Dispersion**

Model Problem Dissipation and Dispersion with and periodic boundary conditions. Solution

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**Dissipation and Dispersion**

Model Problem Dissipation and Dispersion represents Decay dissipation relation represents Propagation dispersion relation For exact solution of no dissipation (constant) no dispersion

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**Dissipation and Dispersion**

Modified Equation Dissipation and Dispersion First Order Upwind Lax-Wendroff Beam-Warming

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**Dissipation and Dispersion**

Modified Equation Dissipation and Dispersion 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

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**Dissipation and Dispersion**

Examples Dissipation and Dispersion First Order Upwind

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**Dissipation and Dispersion**

Examples Dissipation and Dispersion Lax-Wendroff (left) vs. Beam-Warming (right)

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**Dissipation and Dispersion**

Exact Discrete Relations Dissipation and Dispersion For the exact solution , and For the discrete solution

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