Two ADI compact difference methods for variable-exponent diffusion wave equations

Hao Zhang

doi:10.1016/j.apnum.2025.12.003

Two ADI compact difference methods for variable-exponent diffusion wave equations

2026-01-01

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DOI：10.1016/j.apnum.2025.12.003

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Abstract

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In this work, we study two-dimensional diffusion-wave equations with variable exponent, modeling mechanical diffusive wave propagation in viscoelastic media with spatially varying properties. We first transform the diffusion-wave model into an equivalent form via the convolution method. Two time discretization strategies are then applied to approximate each term in the transformed equation, yielding two fully discrete schemes based on a spatial compact finite difference method. To reduce computational cost, the alternating direction implicit (ADI) technique is employed. We prove that both ADI compact schemes are unconditionally stable and convergent. The error estimates established under reasonable regularity assumption, state that the first scheme achieves a(0)-order accuracy in time and fourth-order accuracy in space, while the second scheme attains second-order accuracy in time and fourth-order accuracy in space. Numerical experiments confirm the theoretical predictions and demonstrate the efficiency of the proposed methods.

Keywords:

Two-dimensional diffusion wave

Variable exponent

Averaged PI rule

ADI compact difference

Stability and convergence

Two ADI compact difference methods for variable-exponent diffusion wave equations

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