- The Test Problem and DOPRI54
- One step methods
- Multi step methods
- Global and local errors
- Error estimation
- The Van der Pol System
- Design your own Explicit Runge-Kutta Method
- Order conditions, coefficients for the error estimator and the Butcher tableau
- Testing on the test equation
- Verifying the order
- Stability plot
- Testing on the Van der Pol problem and comparison with ode15s
- Step size controller
- ESDIRK23
- Implementation with fixed step size
- Testing on Van der Pol problem and comparison with our designed ERK
- Stability region, A and L-stability, practical implications
- Implementation with variable step size and testing on Van der Pol problem
- 2 point Boundary Value Problems
- Newton’s method
- Single shooting
- Sensitivity analysis and non-convergence
- 9-point Laplacian
- 9-point stencil
- Iterative solvers in 2D
- Matrix free 5-point Laplacian
- Relaxation and smoothing
- Coarsing / interpolating
- Recursion
- Pure parabolic model problem 2
- Matlab implementation of the theta-scheme
- Scheme analysis
- Numerical evaluation
- Purely hyperbolic model problem
- Stability of the FTBS scheme
- Matlab implementation of FTBS and convergence
- Dispersion and diffusion
- Non-linear advection-diffusion equation
- Stable numerical scheme
- Matlab implementation and convergence
- Testing
- Approaches to maintain accuracy
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