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What makes a differential equations nonlinear?
Examples on linear and nonlinear differential equations
Introduction of basic concepts
The scaled logistic ODE
Linearization by explicit time discretization
An implicit method: Backward Euler discretization
Detour: new notation
Exact solution of quadratic nonlinear equations
How do we pick the right solution in this case?
Linearization
Picard iteration
The algorithm of Picard iteration
The algorithm of Picard iteration with classical math notation
Stopping criteria
A single Picard iteration
Implicit Crank-Nicolson discretization
Linearization by a geometric mean
Newton's method
Newton's method with an iteration index
Using Newton's method on the logistic ODE
Using Newton's method on the logistic ODE with typical math notation
Relaxation may improve the convergence
Implementation; part 1
Implementation; part 2
Implementation; part 3
Experiments: accuracy of iteration methods
Experiments: number of iterations
The effect of relaxation can potentially be great!
Generalization to a general nonlinear ODE
Explicit time discretization
Backward Euler discretization
Picard iteration for Backward Euler scheme
Manual linearization for a given \( f(u,t) \)
Computational experiments with partially implicit treatment of \( f \)
Newton's method for Backward Euler scheme
Crank-Nicolson discretization
Picard and Newton iteration in the Crank-Nicolson case
Systems of ODEs
A Backward Euler scheme for the vector ODE \( u^{\prime}=f(u,t) \)
Example: Crank-Nicolson scheme for the oscillating pendulum model
The nonlinear \( 2\times 2 \) system
Systems of nonlinear algebraic equations
Notation for general systems of algebraic equations
Picard iteration
Algorithm for relaxed Picard iteration
Newton's method for \( F(u)=0 \)
Algorithm for Newton's method
Newton's method for \( A(u)u=b(u) \)
Comparison of Newton and Picard iteration
Combined Picard-Newton algorithm
Stopping criteria
Combination of absolute and relative stopping criteria
Example: A nonlinear ODE model from epidemiology
Implicit time discretization
A Picard iteration
Newton's method
Actually no need to bother with nonlinear algebraic equations for this particular model...
Linearization at the differential equation level
PDE problem
Explicit time integration
Backward Euler scheme
Picard iteration for Backward Euler scheme
Picard iteration with alternative notation
Backward Euler scheme and Newton's method
Calculation details of Newton's method at the PDE level
Calculation details of Newton's method at the PDE level
Result of Newton's method at the PDE level
Similarity with Picard iteration
Using new notation for implementation
Combined Picard and Newton formulation
Crank-Nicolson discretization
Arithmetic means: which variant is best?
Solution of nonlinear equations in the Crank-Nicolson scheme
Discretization of 1D stationary nonlinear differential equations
Relevance of this stationary 1D problem
Finite difference discretizations
Finite difference scheme
Boundary conditions
The structure of the equation system
The equation for the Neumann boundary condition
The equation for the Dirichlet boundary condition
Picard iteration
Details: without Dirichlet condition equation
Details: with Dirichlet condition equation
Newton's method; Jacobian (1)
Newton's method; Jacobian (2)
Newton's method; nonlinear equations at the end points
Galerkin-type discretizations
The nonlinear algebraic equations
Fundamental integration problem: how to deal with \( \int f(\sum_kc_k\baspsi_k)\baspsi_idx \) for unknown \( c_k \)?
We choose \( \baspsi_i \) as finite element basis functions
The group finite element method
What is the point with the group finite element method?
Simplified problem for symbolic calculations
Integrating very simple nonlinear functions results in complicated expressions in the finite element method
Application of the group finite element method
Lumping the mass matrix gives finite difference form
Alternative: evaluation of finite element terms at nodes gives great simplifications
Numerical integration of nonlinear terms
Finite elements for a variable coefficient Laplace term
Numerical integration at the nodes
Summary of finite element vs finite difference nonlinear algebraic equations
Real computations utilize accurate numerical integration
Picard iteration defined from the variational form
The linear system in Picard iteration
The equations in Newton's method
Useful formulas for computing the Jacobian
Computing the Jacobian
Computations in a reference cell \( [-1,1] \)
How to handle Dirichlet conditions in Newton's method
Multi-dimensional PDE problems
Backward Euler and variational form
Nonlinear algebraic equations arising from the variational form
A note on our notation and the different meanings of \( u \) (1)
A note on our notation and the different meanings of \( u \) (2)
Newton's method (1)
Newton's method (2)
Non-homogeneous Neumann conditions
Robin condition
Finite difference discretization in a 2D problem
Picard iteration
Newton's method: the nonlinear algebraic equations
Newton's method: the Jacobian and its sparsity
Newton's method: details of the Jacobian
Good exercise at this point: \( J_{i,j,i,j} \)
Continuation methods
Continuation method: solve difficult problem as a sequence of simpler problems
Example on a continuation method
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