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Shanghai Jiao Tong University
Example: 1D convection-diffusion equation
![Page 2: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/2.jpg)
Shanghai Jiao Tong University
Discretized convection-diffusion equation
![Page 3: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/3.jpg)
Shanghai Jiao Tong UniversityExact solution of the difference scheme
![Page 4: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/4.jpg)
Shanghai Jiao Tong UniversityNumerical behavior of the difference scheme
![Page 5: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/5.jpg)
Shanghai Jiao Tong UniversityEvaluation of the central difference scheme
![Page 6: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/6.jpg)
Shanghai Jiao Tong UniversityEvaluation of the upwind difference scheme
![Page 7: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/7.jpg)
Shanghai Jiao Tong University
Finite Element Method
![Page 8: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/8.jpg)
Shanghai Jiao Tong UniversityFinite Element Method
![Page 9: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/9.jpg)
Shanghai Jiao Tong University
1D minimization problem
![Page 10: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/10.jpg)
Shanghai Jiao Tong University
1D minimization problem
![Page 11: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/11.jpg)
Shanghai Jiao Tong University
Du Bois Reymond lemma
![Page 12: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/12.jpg)
Shanghai Jiao Tong University1D Poisson equation
![Page 13: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/13.jpg)
Shanghai Jiao Tong University
2D Poisson equation
![Page 14: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/14.jpg)
Shanghai Jiao Tong University
2D Poisson equation
![Page 15: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/15.jpg)
Shanghai Jiao Tong University
Rayleigh-Ritz method
![Page 16: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/16.jpg)
Shanghai Jiao Tong University1D Poisson equation
![Page 17: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/17.jpg)
Shanghai Jiao Tong University
Poor choice of the basis functions
![Page 18: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/18.jpg)
Shanghai Jiao Tong University
Finite Element Method
![Page 19: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/19.jpg)
Shanghai Jiao Tong University
Finite Element Method
![Page 20: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/20.jpg)
Shanghai Jiao Tong University1D Poisson equation
![Page 21: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/21.jpg)
Shanghai Jiao Tong University1D Poisson equation
![Page 22: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/22.jpg)
Shanghai Jiao Tong University
Existence of a minimization problem
![Page 23: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/23.jpg)
Shanghai Jiao Tong University1D Poisson equation
![Page 24: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/24.jpg)
Shanghai Jiao Tong UniversityLeast-squares method
![Page 25: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/25.jpg)
Shanghai Jiao Tong University
Weighted residual method
![Page 26: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/26.jpg)
Shanghai Jiao Tong University1D Poisson equation
![Page 27: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/27.jpg)
Shanghai Jiao Tong University2D Poisson equation
![Page 28: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/28.jpg)
Shanghai Jiao Tong University2D Poisson equation
![Page 29: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/29.jpg)
Shanghai Jiao Tong UniversityConstruction of 1D finite elements
![Page 30: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/30.jpg)
Shanghai Jiao Tong UniversityConstruction of 1D finite elements
![Page 31: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/31.jpg)
Shanghai Jiao Tong UniversityConstruction of 1D finite elements
![Page 32: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/32.jpg)
Shanghai Jiao Tong UniversityConstruction of triangular finite elements
![Page 33: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/33.jpg)
Shanghai Jiao Tong UniversityConstruction of triangular finite elements
![Page 34: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/34.jpg)
Shanghai Jiao Tong UniversityConstruction of triangular finite elements
![Page 35: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/35.jpg)
Shanghai Jiao Tong UniversityConstruction of tetrahedral finite elements
![Page 36: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/36.jpg)
Shanghai Jiao Tong UniversityCoordinate and element transformations
![Page 37: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/37.jpg)
Shanghai Jiao Tong UniversityCoordinate and element transformations
![Page 38: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/38.jpg)
Shanghai Jiao Tong UniversityCoordinate and element transformations
![Page 39: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/39.jpg)
Shanghai Jiao Tong UniversityCoordinate and element transformations
![Page 40: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/40.jpg)
Shanghai Jiao Tong University
Construction of qudarilateral finite elements
![Page 41: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/41.jpg)
Shanghai Jiao Tong University
Construction of qudarilateral finite elements
![Page 42: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/42.jpg)
Shanghai Jiao Tong University
Construction of hexahedral finite elements
![Page 43: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/43.jpg)
Shanghai Jiao Tong University
Finite elements matrix assembly
![Page 44: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/44.jpg)
Shanghai Jiao Tong University
1D Poisson equation/linear element
![Page 45: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/45.jpg)
Shanghai Jiao Tong University
1D Poisson equation/linear element
![Page 46: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/46.jpg)
Shanghai Jiao Tong University
1D Poisson equation/linear element
![Page 47: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/47.jpg)
Shanghai Jiao Tong University1D Poisson equation/quadratic element
![Page 48: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/48.jpg)
Shanghai Jiao Tong UniversityNumerical integration for finite element
![Page 49: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/49.jpg)
Shanghai Jiao Tong UniversityStorage of space matrices
![Page 50: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/50.jpg)
Shanghai Jiao Tong University
Discretization in Time
![Page 51: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/51.jpg)
Shanghai Jiao Tong UniversityIBV problems
![Page 52: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/52.jpg)
Shanghai Jiao Tong UniversitySpace-time discretization
![Page 53: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/53.jpg)
Shanghai Jiao Tong UniversityStandard time-stepping schemes
![Page 54: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/54.jpg)
Shanghai Jiao Tong University
Properties of time-stepping schemes
![Page 55: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/55.jpg)
Shanghai Jiao Tong University
Explicit vs. implicit time discretization
![Page 56: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/56.jpg)
Shanghai Jiao Tong University
1D convection-diffusion equation
![Page 57: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/57.jpg)
Shanghai Jiao Tong University
1D convection-diffusion equation
![Page 58: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/58.jpg)
Shanghai Jiao Tong University
1D convection-diffusion equation
![Page 59: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/59.jpg)
Shanghai Jiao Tong University
Fractional-step ɵ-scheme
![Page 60: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/60.jpg)
Shanghai Jiao Tong UniversityPredictor-corrector and multipoint methods
![Page 61: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/61.jpg)
Shanghai Jiao Tong UniversityAdams methods
![Page 62: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/62.jpg)
Shanghai Jiao Tong UniversityAdams methods
![Page 63: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/63.jpg)
Shanghai Jiao Tong UniversityRunge-Kutta methods
![Page 64: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/64.jpg)
Shanghai Jiao Tong UniversityRunge-Kutta methods
![Page 65: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/65.jpg)
Shanghai Jiao Tong University
Automatic time step control
![Page 66: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/66.jpg)
Shanghai Jiao Tong UniversityAutomatic time step control
![Page 67: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/67.jpg)
Shanghai Jiao Tong UniversityPractical implementation
![Page 68: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/68.jpg)
Shanghai Jiao Tong University
Evolutionary PID controller
![Page 69: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/69.jpg)
Shanghai Jiao Tong University
Pseudo time-stepping
![Page 70: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/70.jpg)
Shanghai Jiao Tong University
Properties of numerical methods
![Page 71: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/71.jpg)
Shanghai Jiao Tong University
Properties of numerical methods
![Page 72: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/72.jpg)
Shanghai Jiao Tong UniversityConsistency
![Page 73: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/73.jpg)
Shanghai Jiao Tong University
Stability
![Page 74: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/74.jpg)
Shanghai Jiao Tong UniversityMatrix method for stability analysis
![Page 75: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/75.jpg)
Shanghai Jiao Tong UniversityMatrix method for stability analysis
![Page 76: Example: 1D convection-diffusion equation · Discretized convection-diffusion equation. ... Exact solution of the difference scheme. Shanghai Jiao Tong University ... Shanghai Jiao](https://reader031.vdocuments.us/reader031/viewer/2022013023/6052929d0331104d9e17c053/html5/thumbnails/76.jpg)
Shanghai Jiao Tong UniversityVon Neumann’s stability anaylsis
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Shanghai Jiao Tong UniversityVon Neumann’s stability anaylsis
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Shanghai Jiao Tong University1D pure convection equation
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Shanghai Jiao Tong University1D pure convection equation
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Shanghai Jiao Tong University1D pure convection equation
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Shanghai Jiao Tong UniversitySpectral analysis of numerical errors
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Shanghai Jiao Tong UniversityConvergency
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Shanghai Jiao Tong UniversityConservation
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Shanghai Jiao Tong UniversityDiscrete conservation
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Shanghai Jiao Tong UniversityBoundedness