A Higher-Order Multiscale Method for the Wave Equation

TL;DR

This paper tackles efficient simulation of the acoustic wave equation in highly oscillatory media by introducing a higher-order multiscale discretization that integrates a $p$-LOD spatial approximation with a $\theta$-based time-stepping scheme. The method constructs a localized multiscale space $\tilde{U}_H^\ell$ using extended bubble corrections and couples it with a higher-order time integrator, enabling a priori error estimates in the energy norm. Theoretical findings show that, for rough $L^\infty$ coefficients, spatial convergence is capped at $r\le 2$, though increasing the polynomial degree $p$ reduces the error size; under additional regularity or smoother coefficients, higher-order spatial convergence up to $r\le p+2$ can be achieved, with temporal order $s=2$ or $4$ depending on $\theta$ (notably $\theta=1/12$ yields the highest time accuracy). Numerical experiments with oscillatory coefficients confirm the theory and illustrate that higher $p$ improves accuracy for a given number of degrees of freedom, while smooth coefficients enable higher rates, validating the method’s efficiency for multiscale wave simulations and guiding future refinement toward truly high-order schemes under minimal assumptions.

Abstract

In this paper we propose a multiscale method for the acoustic wave equation in highly oscillatory media. We use a higher-order extension of the localized orthogonal decomposition method combined with a higher-order time stepping scheme and present rigorous a-priori error estimates in the energy-induced norm. We find that in the very general setting without additional assumptions on the coefficient beyond boundedness, arbitrary orders of convergence cannot be expected but that increasing the polynomial degree may still considerably reduce the size of the error. Under additional regularity assumptions, higher orders can be obtained as well. Numerical examples are presented that confirm the theoretical results.

Figures (6)

• Figure 2.1: Plots of different basis functions: three Legendre polynomials $\Lambda_{K,1}$ (top left), the corresponding bubble functions $b_{K,1}$ (top right), the functions $\iota_K$ (bottom left), $\nu_K$ (bottom middle), and the newly created basis function $\iota_K+\nu_K$ (bottom right).
• Figure 4.1: Coefficient $A_1$ (left) with values between $1$ and $10$ and coefficient $A_2$ (right) with values between $1$ and $9$. Both coefficients vary on the scale $\varepsilon=2^{-6}$.
• Figure 4.2: Relative $|\cdot|_a$-errors for the coefficient $A_1$ with respect to the coarse mesh size $H$ (left) and the degrees of freedom $\mathcal{N}$ (right) for different polynomial degrees $p$ and localization parameters $\ell$.
• Figure 4.3: Relative $|\cdot|_a$-errors for the second example with respect to the coarse mesh size $H$ (left) and the number of degrees of freedom $\mathcal{N}$ (right) for different polynomial degrees $p$ and localization parameters $\ell$.
• Figure 4.4: Relative $L^2$-errors for the second example for $\theta=\frac{1}{12}$ (left) and relative $|\cdot|_a$-errors for the second example for $\theta=\frac{1}{4}$ compared to $\theta=\frac{1}{12}$ for $p=2$ (right) with time step sizes dependent on $H$ and $h$, respectively.

Theorems & Definitions (23)

• Remark 2.1
• Lemma 2.2
• Lemma 3.1: Energy conservation
• proof
• Theorem 3.2: Stability
• proof
• Remark 3.3
• Remark 3.5
• Theorem 3.6: Error of the $p$-LOD-$\theta$ method
• Remark 3.7