Numerical Wave Equation

PDEs · Axiom Academy

EXAMPLE Numerical Wave Equation Step-by-step solutions for explicit finite difference schemes Excellent work! You've mastered the explicit finite difference scheme for the wave equation. Here's what we learned: Central Differences: Using central differences for both spatial and temporal derivatives gives second-order accuracy in space and time. Explicit Time-Stepping: The scheme is explicit because we can directly compute u j n+1 from known values at previous time levels without solving a system of equations. CFL Condition: For stability, the numerical domain of dependence must contain the physical domain of dependence. This requires = c t/ x 1 . Three Time Levels: The scheme requires values at two previous time levels (n and n-1) to compute the next time level (n+1). Special treatment is needed for the first time step. Initial Pulse Propagation: With proper initial conditions and CFL number, the scheme accurately captures wave propagation, maintaining pulse shape as it travels through the domain. Boundary Conditions: Fixed boundary conditions (u=0 at endpoints) are enforced at each time step, causing wave reflection at boundaries. This explicit scheme is simple to implement but requires small time steps for stability. For larger time steps, consider implicit schemes like Crank-Nicolson, which are unconditionally stable but require solving a tridiagonal system at each time step.

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