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

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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 Characteristics General solution

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First Order Wave Equation Solution

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First Order Wave Equation Solution

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First Order Wave Equation Stability

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Model Problem Initial condition: Periodic Boundary conditions: constant

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Model Problem Example Periodic Solution (U>0)

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

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Finite Difference Solution Discretization

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

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

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

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

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

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

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Finite Difference Solution First Order Upwind Scheme Example

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Convergence Definition The finite difference algorithm converges if For any initial condition.

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Consistency Definition For all smooth functions when. The difference scheme, is consistent with the differential equation if:

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Consistency First Order Upwind Scheme Difference operator Differential operator

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

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Stability First Order Upwind Scheme

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Stability First Order Upwind Scheme

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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 ( first order in, )

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

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

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CFL Condition Domains of dependence 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.

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CFL Condition Domains of dependence First Order Upwind Scheme AnalyticalNumerical ( 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 Condition CFL Theorem StableUnstable

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

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

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Fourier Analysis Continuous Problem Wave Equation

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

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

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

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

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

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

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

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

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

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

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Lax-Wendroff Scheme Spatial Approximation Approximate spatial derivatives

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

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Lax-Wendroff Scheme Interpretation Use Quadratic Interpolation

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

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Lax-Wendroff Scheme Analysis Truncation Error Consistency Insert exact solution into difference scheme

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Lax-Wendroff Scheme Analysis Stability Stability if:

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Lax-Wendroff Scheme Analysis Convergence Consistency: Stability: Convergence and are constants independent of

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Lax-Wendroff Scheme Domains of Dependence AnalyticalNumerical

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Lax-Wendroff Scheme CFL Condition StableUnstable

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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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Beam-Warming Scheme Derivation Use Quadratic Interpolation

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Beam-Warming Scheme Consistency and Stability 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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Method of Lines Spatial Discretization Central difference … (for example) or, in vector form,

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

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

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

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Method of Lines Spatial Discretization Fourier Analysis …

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

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

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Method of Lines Fourier Stability Analysis Fourier Transform

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Method of Lines Fourier Stability Analysis Application factor with Stability

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

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

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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

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Method of Lines Methods for ODE s Model equation: complex- valued Discretization EF EB CN

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

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

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Method of Lines Methods for ODE s … Absolute Stability Diagrams

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

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

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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

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Method of Lines Methods for ODE s … Application to the wave equation … Explicit Time Stepping Scheme Predictor/Corrector

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Method of Lines Methods for ODE s … Application to the wave equation … Explicit Time Stepping Scheme 4 Stage Runge-Kutta

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Method of Lines Methods for ODE s … 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 … Application to the wave equation … Adding Artificial Viscosity For each Fourier mode θ, Additional Term

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

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

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

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Dissipation and Dispersion Model Problem 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 First Order Upwind Lax-Wendroff Beam-Warming

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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

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Dissipation and Dispersion Examples First Order Upwind

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

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

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