Fast Spectral Simulation of One-Dimensional Nonlinear Fractional Waves in Attenuating Media
Abstract
Wave equations with Caputo fractional damping model power-law attenuation in rocks, but their numerical solution is costly because of the non-local memory, and stability proofs for nonlinear variants are error-prone. We study a one-dimensional wave equation with Caputo damping of order α∈(0,1) and a cubic nonlinearity, discretised by Legendre–Gauss–Lobatto (LGL) collocation in space and a linearly implicit three-level scheme with the L1 formula in time. The memory is compressed by a sum-of-exponentials (SOE) approximation with positive weights, which reduces the cost of the history term from O(N2t) to O(NeNt) with Ne growing logarithmically. By a discrete energy argument that does not use a sign of the discrete memory form, we prove stability under a time-step restriction independent of the polynomial degree and an error bound of order τ2-α+N1-m+εSOE; the energy argument is machine-checked in Lean 4. Experiments confirm the temporal rates, spectral accuracy in space and a speed-up of the memory evaluation of up to 60 times. In a seismic application with the damping calibrated from a prescribed quality factor, the quality factor recovered from simulated seismograms agrees with the dispersion relation to within 0.5%. An extension to Caputo’s constitutive model reproduces a nearly constant quality factor.
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