dev.nm.analysis.differentialequation.pde.finitedifference.hyperbolic.dim1

Class ExplicitCentralDifference1D

java.lang.Object

dev.nm.analysis.differentialequation.pde.finitedifference.hyperbolic.dim1.ExplicitCentralDifference1D

All Implemented Interfaces:

PDESolver

public class ExplicitCentralDifference1D
extends Object
implements PDESolver

This explicit central difference method is a numerical technique for solving the one-dimensional wave equation by the following explicit three-point central difference equation. \[ \frac{u^{k+1}_j - 2u^k_j + u^{k-1}_j}{\Delta t^2} = \beta \frac{u^{k}_{j+1} - 2u^k_j + u^{k}_{j-1}}{\Delta x^2} \] where \(u^k_j = u(t_k, x_j)\) is the estimate at \((k, j)\) in the grid.

This method is NOT unconditionally stable. Specifically, it is up to the user to ensure that: \[ \Delta t^2 \leq \frac{\Delta x^2}{\beta} \] where \(\Delta t = \frac{T}{m}\) and \(\Delta x = \frac{a}{n+1}\), by specifying the grid resolution parameters m and n.

This is a second-order method with a truncation error of order \(O(\Delta t^2 + \Delta x^2)\).

Constructor Summary

Constructors

Constructor and Description
ExplicitCentralDifference1D()

Method Summary

Modifier and Type	Method and Description
PDESolutionTimeSpaceGrid1D	solve(WaveEquation1D pde, int m, int n) Solve an one-dimensional wave equation, with the resolution parameters of the solution grid.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

ExplicitCentralDifference1D

public ExplicitCentralDifference1D()

Method Detail

solve

public PDESolutionTimeSpaceGrid1D solve(WaveEquation1D pde,
                                        int m,
                                        int n)

Solve an one-dimensional wave equation, with the resolution parameters of the solution grid.

Parameters:

pde - the wave equation problem

m - the number of grid points along the time-axis

n - the number of grid points along the space-axis

Returns:

the solution grid