example: 1d convection-diffusion equation · discretized convection-diffusion equation. ... exact...

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Example: 1D convection-diffusion equation

Shanghai Jiao Tong University

Discretized convection-diffusion equation

Shanghai Jiao Tong UniversityExact solution of the difference scheme

Shanghai Jiao Tong UniversityNumerical behavior of the difference scheme

Shanghai Jiao Tong UniversityEvaluation of the central difference scheme

Shanghai Jiao Tong UniversityEvaluation of the upwind difference scheme

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Finite Element Method

Shanghai Jiao Tong UniversityFinite Element Method

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1D minimization problem

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1D minimization problem

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Du Bois Reymond lemma

Shanghai Jiao Tong University1D Poisson equation

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2D Poisson equation

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2D Poisson equation

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Rayleigh-Ritz method

Shanghai Jiao Tong University1D Poisson equation

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Poor choice of the basis functions

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Finite Element Method

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Finite Element Method

Shanghai Jiao Tong University1D Poisson equation

Shanghai Jiao Tong University1D Poisson equation

Shanghai Jiao Tong University

Existence of a minimization problem

Shanghai Jiao Tong University1D Poisson equation

Shanghai Jiao Tong UniversityLeast-squares method

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Weighted residual method

Shanghai Jiao Tong University1D Poisson equation

Shanghai Jiao Tong University2D Poisson equation

Shanghai Jiao Tong University2D Poisson equation

Shanghai Jiao Tong UniversityConstruction of 1D finite elements

Shanghai Jiao Tong UniversityConstruction of 1D finite elements

Shanghai Jiao Tong UniversityConstruction of 1D finite elements

Shanghai Jiao Tong UniversityConstruction of triangular finite elements

Shanghai Jiao Tong UniversityConstruction of triangular finite elements

Shanghai Jiao Tong UniversityConstruction of triangular finite elements

Shanghai Jiao Tong UniversityConstruction of tetrahedral finite elements

Shanghai Jiao Tong UniversityCoordinate and element transformations

Shanghai Jiao Tong UniversityCoordinate and element transformations

Shanghai Jiao Tong UniversityCoordinate and element transformations

Shanghai Jiao Tong UniversityCoordinate and element transformations

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Construction of qudarilateral finite elements

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Construction of qudarilateral finite elements

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Construction of hexahedral finite elements

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Finite elements matrix assembly

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1D Poisson equation/linear element

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1D Poisson equation/linear element

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1D Poisson equation/linear element

Shanghai Jiao Tong University1D Poisson equation/quadratic element

Shanghai Jiao Tong UniversityNumerical integration for finite element

Shanghai Jiao Tong UniversityStorage of space matrices

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Discretization in Time

Shanghai Jiao Tong UniversityIBV problems

Shanghai Jiao Tong UniversitySpace-time discretization

Shanghai Jiao Tong UniversityStandard time-stepping schemes

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Properties of time-stepping schemes

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Explicit vs. implicit time discretization

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1D convection-diffusion equation

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1D convection-diffusion equation

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1D convection-diffusion equation

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Fractional-step ɵ-scheme

Shanghai Jiao Tong UniversityPredictor-corrector and multipoint methods

Shanghai Jiao Tong UniversityAdams methods

Shanghai Jiao Tong UniversityAdams methods

Shanghai Jiao Tong UniversityRunge-Kutta methods

Shanghai Jiao Tong UniversityRunge-Kutta methods

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Automatic time step control

Shanghai Jiao Tong UniversityAutomatic time step control

Shanghai Jiao Tong UniversityPractical implementation

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Evolutionary PID controller

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Pseudo time-stepping

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Properties of numerical methods

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Properties of numerical methods

Shanghai Jiao Tong UniversityConsistency

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Stability

Shanghai Jiao Tong UniversityMatrix method for stability analysis

Shanghai Jiao Tong UniversityMatrix method for stability analysis

Shanghai Jiao Tong UniversityVon Neumann’s stability anaylsis

Shanghai Jiao Tong UniversityVon Neumann’s stability anaylsis

Shanghai Jiao Tong University1D pure convection equation

Shanghai Jiao Tong University1D pure convection equation

Shanghai Jiao Tong University1D pure convection equation

Shanghai Jiao Tong UniversitySpectral analysis of numerical errors

Shanghai Jiao Tong UniversityConvergency

Shanghai Jiao Tong UniversityConservation

Shanghai Jiao Tong UniversityDiscrete conservation

Shanghai Jiao Tong UniversityBoundedness