| Class | Description |
|---|---|
| ExplicitCentralDifference2D |
This explicit central difference method is a numerical technique for solving
the two-dimensional wave equation by the following explicit
three-point central difference equation.
\[
\frac{u^{k+1}_{ij} - 2u^k_{ij} + u^{k-1}_{ij}}{\Delta t^2}
= \beta \left( \frac{u^{k}_{i+1,j} - 2u^k_{ij} + u^{k}_{i-1,j}}{\Delta x^2}
+ \frac{u^{k}_{i,j+1} - 2u^k_{ij} + u^{k}_{i,j-1}}{\Delta y^2}
\right)
\]
where \(u^k_{ij} = u(t_k,x_i,y_j)\) is the estimate at \((k, i, j)\) in the grid.
|
| WaveEquation2D |
A two-dimensional wave equation is a hyperbolic PDE that takes the following form.
\[
\frac{\partial^2 u}{\partial t^2}
= \beta (\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2})
\]
with fixed boundary condition, that is,
\[
u(t, 0, 0) = u(t, a, 0) = u(t, 0, b) = u(t, a, b) = 0
\]
and two initial conditions, namely \(u\) and its first time derivative \(u_t\):
\[
u(0, x, y) = f(x, y), 0 < x < a, 0 < y < b \\
u_t(0, x, y) = g(x, y), 0 < x < a, 0 < y < b
\]
|
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