Solving nonlinear ODE and PDE problems¶
Contents:
- Solving nonlinear ODE and PDE problems
- Introduction of basic concepts
- Linear versus nonlinear equations
- A simple model problem
- Linearization by explicit time discretization
- Exact solution of nonlinear algebraic equations
- Linearization
- Picard iteration
- Linearization by a geometric mean
- Newton’s method
- Relaxation
- Implementation and experiments
- Generalization to a general nonlinear ODE
- Systems of ODEs
- Systems of nonlinear algebraic equations
- Linearization at the differential equation level
- Discretization of 1D stationary nonlinear differential equations
- Multi-dimensional PDE problems
- Exercises
- Problem 1: Determine if equations are nonlinear or not
- Exercise 2: Derive and investigate a generalized logistic model
- Problem 3: Experience the behavior of Newton’s method
- Problem 4: Compute the Jacobian of a \(2\times 2\) system
- Problem 5: Solve nonlinear equations arising from a vibration ODE
- Exercise 6: Find the truncation error of arithmetic mean of products
- Problem 7: Newton’s method for linear problems
- Exercise 8: Discretize a 1D problem with a nonlinear coefficient
- Exercise 9: Linearize a 1D problem with a nonlinear coefficient
- Problem 10: Finite differences for the 1D Bratu problem
- Problem 11: Integrate functions of finite element expansions
- Problem 12: Finite elements for the 1D Bratu problem
- Exercise 13: Discretize a nonlinear 1D heat conduction PDE by finite differences
- Exercise 14: Use different symbols for different approximations of the solution
- Exercise 15: Derive Picard and Newton systems from a variational form
- Exercise 16: Derive algebraic equations for nonlinear 1D heat conduction
- Exercise 17: Differentiate a highly nonlinear term
- Exercise 18: Crank-Nicolson for a nonlinear 3D diffusion equation
- Exercise 19: Find the sparsity of the Jacobian
- Problem 20: Investigate a 1D problem with a continuation method
- Bibliography
- Appendix: Symbolic nonlinear finite element equations
