Solving nonlinear ODE and PDE problems

Contents:

• Solving nonlinear ODE and PDE problems
• Introduction of basic concepts
  • Linearization by explicit time discretization
  • Exact solution of nonlinear equations
  • Linearization
  • Picard iteration (1)
    • Stopping criteria (1)
    • A single Picard iteration
  • Linearization by a geometric mean
  • Newton’s method (1)
  • Relaxation
  • Implementation and experiments
  • Generalization to a general nonlinear ODE
    • Explicit time discretization
    • Backward Euler discretization
    • Crank-Nicolson discretization (1)
  • Systems of ODEs
• Systems of nonlinear algebraic equations
  • Picard iteration (2)
  • Newton’s method (2)
  • Stopping criteria (2)
  • Example: A nonlinear ODE model from epidemiology
    • Implicit time discretization
    • A Picard iteration
    • Newton’s method (3)
• Linearization at the differential equation level
  • Explicit time integration
  • Backward Euler scheme and Picard iteration
  • Backward Euler scheme and Newton’s method
    • Linearization via Taylor expansions
    • Similarity with Picard iteration
    • Implementation
    • Derivation with alternative notation
  • Crank-Nicolson discretization (2)
• Discretization of 1D stationary nonlinear differential equations
  • Finite difference discretizations
  • Solution of algebraic equations
    • The structure of the equation system
    • Picard iteration (3)
    • Newton’s method (4)
  • Galerkin-type discretizations
    • Fundamental integration problem
  • Finite element basis functions
  • The group finite element method
    • Finite element approximation of functions of \(u\)
    • Simplified problem
    • Integrating nonlinear functions
    • Application of the group finite element method
  • Numerical integration of nonlinear terms
  • Finite element discretization of a variable coefficient Laplace term
    • Group finite element method
    • Numerical integration at the nodes
  • Picard iteration defined from the variational form
  • Newton’s method defined from the variational form
    • Dirichlet conditions
• Multi-dimensional PDE problems
  • Finite element discretization
    • Non-homogeneous Neumann conditions
    • Robin conditions
  • Finite difference discretization
    • Picard iteration (4)
    • Newton’s method (5)
  • Continuation methods
• Exercises
  • Problem 1: Determine if equations are nonlinear or not
  • Problem 2: Experience the behavior of Newton’s method
  • Problem 3: Compute the Jacobian of a \(2\times 2\) system
  • Problem 4: Solve nonlinear equations arising from a vibration ODE
  • Exercise 5: Find the truncation error of arithmetic mean of products
  • Problem 6: Newton’s method for linear problems
  • Exercise 7: Discretize a 1D problem with a nonlinear coefficient
  • Exercise 8: Linearize a 1D problem with a nonlinear coefficient
  • Problem 9: Finite differences for the 1D Bratu problem
  • Problem 10: Integrate functions of finite element expansions
  • Problem 11: Finite elements for the 1D Bratu problem
  • Exercise 12: Discretize a nonlinear 1D heat conduction PDE by finite differences
  • Exercise 13: Use different symbols for different approximations of the solution
  • Exercise 14: Derive Picard and Newton systems from a variational form
  • Exercise 15: Derive algebraic equations for nonlinear 1D heat conduction
  • Exercise 16: Differentiate a highly nonlinear term
  • Exercise 17: Crank-Nicolson for a nonlinear 3D diffusion equation
  • Exercise 18: Find the sparsity of the Jacobian
  • Problem 19: Investigate a 1D problem with a continuation method
• Bibliography

Indices and tables

• Index
• Module Index
• Search Page