微分方程数值解速通

Initial Value Problems

Problem formulation:

Numerical Differentiation Based Methods

1. Grids
2. Evaluate the differential equation at the grid points
3. Replace the derivative by a difference quotient where is the truncation error. The equation becomes
4. Drop the truncation error term And the initial condition gives
5. Error analysis of . The method is first order accurate.

Common Numerical Differentiation Formulas

Finite difference methods for solving an IVP

Stability

Use the model problem

• A-stable: , is bounded?
• L-stable: For all , , is bounded?
• Zero-stable: Let , , is bounded?

Numerical Integration Based Methods

Numerical Intergration Formulas

1. Grids
2. Integrate the differential equation over the interval That is
3. Replace the integral by a quadrature formula
4. Drop the error term

Other notes:

• The idea of Runge-Kutta methods is to replace future values in implicit methods by values predicted by an explicit method.
• Extension and ghost points are used to handle boundary conditions, by adding extra points to the grid.
• Conservative methods are used to preserve the total energy of the system.

Two-Point Boundary Value Problems

Problem formulation:

Using the five-step finite difference method, we will eventually get a system of linear equations to solve. Constant boundary conditions will give a tridiagonal system:

where

The tridiagonal matrix can be solved in operations. To discuss the existence and uniqueness of the solution, we have these theorems:

• Theorem 1: is invertible when either
  • is strictly diagonally dominant, or
  • is duagonally dominant, and , .
• Theroem 2: If are continuous on , then the solution of the BVP exists and is unique when the step size satisfies
• Theorem 3 (Error Analysis): When the conditions of Theorem 2 are satisfied, the error of the finite difference method is where is a constant (, ), is the truncation error, and are the boundary errors.

Diffusion Problems

Problem formulation: The inhomogeneous heat equation

with initial condition and boundary conditions and .

5-step explicit finite difference method will result in a matrix form of

where is a symmetric tridiagonal matrix. FDE form:

where . To analyze the stability, sbustitute with

and deduce from the FDE form. The method is stable if (regardless of ).

Useful identities for stability analysis

• A-Stability: The method is A-stable if for all .
• L-Stability: The method is L-stable if as .

Advection Problems

Problem formulation: The advection equation

with initial condition and periodic boundary conditions .

CFL condition: The numerical domain of dependence must contain the physical domain of dependence.

Numerical Wave Propagation

Problem formulation: The wave equation

with zero boundary conditions and initial conditions and .

Solution properties: Substitute the plane wave solution

into the wave equation, and extract the coefficients of to get the dispersion relation. Transform the dispersion relation to obtain the phase velocity,

and the group velocity,

• Dispersive: depends on .
• Dissipative: .

Elliptic Problems

Problem formulation: The Laplace equation

with given boundary conditions.

Key points:

• Banded matrices can be used to solve the Laplace equation.
• Positive definite matrices (Geršgorin circle theorem)
• Iterative methods

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