First-Order Elastic Wave Equation Staggered Grid Finite Difference Method

Fundamentals of Elastic Wave Equation Numerical Simulation

In geophysical exploration and seismic wave propagation research, numerical simulation of first-order elastic wave equations serves as a core methodology. It mathematically describes wave propagation patterns in various media, while the staggered grid finite difference method is widely adopted due to its computational efficiency and numerical stability.

### Principle of Staggered Grid Finite Difference Method The core concept of staggered grids involves spatially and temporally displacing wavefield variables (such as velocity components and stress components) to reduce numerical dispersion and enhance calculation accuracy. Specifically: Velocity components (e.g., v_x, v_z) are placed at half-integer grid points Stress components (e.g., σ_xx, σ_zz, σ_xz) are placed at integer grid points In code implementation, this requires separate memory allocation for velocity and stress arrays with shifted indexing. The differential calculations effectively utilize neighboring point information through interleaved addressing, significantly improving numerical stability.

### Second-Order Time and Fourth-Order Spatial Difference Scheme Second-order time differencing (e.g., central difference) ensures temporal accuracy, while fourth-order spatial differencing effectively suppresses high-frequency numerical noise. This combination balances computational precision and cost, making it suitable for most seismic wave simulation scenarios. Algorithm implementation typically involves: - Time update using leapfrog scheme: v(t+Δt) = v(t) + Δt * stress_gradient - Spatial derivatives calculated with fourth-order stencils: ∂σ/∂x ≈ [-σ(i+2) + 8σ(i+1) - 8σ(i-1) + σ(i-2)] / (12Δx)

### Boundary Condition Handling Due to finite computational domains, boundary reflections can contaminate simulation results. Common boundary treatment methods include: Absorbing Boundary Conditions (ABC): Such as PML (Perfectly Matched Layer) that gradually absorbs wavefield energy in boundary layers to reduce artificial reflections. Rigid boundaries: Directly setting fixed values, though this may generate strong reflections. Code implementation of PML typically involves additional damping terms in the governing equations and special treatment of boundary region arrays.

### Wavefield Snapshot Analysis Wavefield snapshots are crucial for visualizing wave propagation processes. By outputting wavefield distributions at fixed time intervals, one can directly observe wave propagation, reflection, and diffraction phenomena. This is essential for validating numerical method correctness and adjusting model parameters. In practice, snapshot generation can be optimized through selective output scheduling and memory-mapped file operations to handle large-scale simulations.

For further optimization, considerations include parallel computing techniques (MPI/OpenMP), higher-order difference schemes, and extensions for anisotropic media simulation.