Finite Difference Explicit Method for Wave Equation

Definition

The Finite Difference Explicit Method for the wave equation is a numerical technique used to approximate the solutions of partial differential equations (PDEs) that describe wave propagation. It replaces continuous derivatives with discrete algebraic differences, allowing the state of a system at a future time step to be calculated directly from known states at current and previous time steps.

Main Content

1. The Governing Wave Equation

• The one-dimensional wave equation is represented as $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$, where $u(x,t)$ is the displacement, $c$ is the wave speed, $x$ is space, and $t$ is time.
• This equation describes physical phenomena such as vibrations in a string or acoustic waves in a pipe.

2. Discretization of the Domain

• We divide the spatial domain into small intervals of length $\Delta x$ and the time domain into intervals of length $\Delta t$.
• We use the notation $u_i^n$ to represent the displacement at position $x_i = i\Delta x$ and time $t_n = n\Delta t$.

3. Central Difference Approximations

• Using Taylor series expansion, the second-order partial derivatives are approximated:
• $\frac{\partial^2 u}{\partial t^2} \approx \frac{u_i^{n+1} - 2u_i^n + u_i^{n-1}}{(\Delta t)^2}$
• $\frac{\partial^2 u}{\partial x^2} \approx \frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{(\Delta x)^2}$

Working / Process

1. Setting up the Explicit Formula

• By substituting the approximations into the wave equation, we isolate the future state term $u_i^{n+1}$.
• The formula becomes: $u_i^{n+1} = 2(1 - r^2)u_i^n + r^2(u_{i+1}^n + u_{i-1}^n) - u_i^{n-1}$, where $r = \frac{c \Delta t}{\Delta x}$ is the Courant number.

2. Implementation of Boundary and Initial Conditions

• Initial conditions specify the displacement $u(x,0)$ and the initial velocity $\frac{\partial u}{\partial t}(x,0)$ at $t=0$.
• Boundary conditions define the values of $u$ at the ends of the domain (e.g., fixed ends where $u=0$).

3. Iterative Computation

• Starting from time step $n=0$, compute $n=1, 2, ...$ sequentially.
• Because it is "explicit," each grid point $u_i^{n+1}$ is computed solely based on values from previous time steps, which is computationally efficient.

Visualizing the Grid Stencil:
Time (n+1)      (u_i^{n+1})  <- Future (To be calculated)
                  |
Time (n)    (u_{i-1}^n)---(u_i^n)---(u_{i+1}^n) <- Known values
                  |
Time (n-1)      (u_i^{n-1})  <- Past value

Advantages / Applications

• Computational Efficiency: It is highly parallelizable and does not require solving large systems of simultaneous linear equations.
• Seismic Modeling: Used extensively in geophysics to simulate how earthquake waves travel through different layers of the Earth.
• Acoustics: Essential in designing concert halls or analyzing sound propagation in complex environments.
• Stability Requirement: The method is only stable if $c \frac{\Delta t}{\Delta x} \leq 1$, known as the CFL condition.

Summary

The Finite Difference Explicit Method is a robust numerical tool for solving wave equations by discretizing space and time into a grid. By applying central difference approximations to the wave equation, we can calculate future displacement values through simple algebraic iterations. The stability of this method is strictly governed by the Courant-Friedrichs-Lewy (CFL) condition, which relates wave speed, time steps, and grid spacing. Important terms include Discretization, CFL Condition, Grid Stencil, and Explicit Integration.