具有连续尺度的声波方程的数值均匀化
Numerical Homogenization of the Acoustic Wave Equations with a Continuum of Scales
AI总结：
本文针对具有有界系数的声波方程，在Cordes型条件下证明了解在调和坐标下的正则性，并提出了一种无需尺度分离或遍历性假设的数值均匀化方法。
摘要：
本文考虑具有异质系数的声波方程的数值均匀化，即当介质的体积模量和密度仅为有界时。我们证明，在Cordes型条件下，解关于调和坐标的二阶导数是$L^2$的（而关于欧几里得坐标是$H^{-1}$），并且解本身属于$L^{\infty}(0,T,H^2(\Omega))$（而关于欧几里得坐标属于$L^{\infty}(0,T,H^1(\Omega))$）。然后，我们提出一种隐式时间步进方法，在粗空间尺度上求解所得的线性系统，并给出该方法的误差估计。由此可知，通过预计算相关的调和坐标，可以在没有尺度分离或遍历性假设的情况下对波动方程进行数值均匀化。
英文摘要：
In this paper, we consider numerical homogenization of acoustic wave equations with heterogeneous coefficients, namely, when the bulk modulus and the density of the medium are only bounded. We show that under a Cordes type condition the second order derivatives of the solution with respect to harmonic coordinates are $L^2$ (instead $H^{-1}$ with respect to Euclidean coordinates) and the solution itself is in $L^{\infty}(0,T,H^2(Ω))$ (instead of $L^{\infty}(0,T,H^1(Ω))$ with respect to Euclidean coordinates). Then, we propose an implicit time stepping method to solve the resulted linear system on coarse spatial scales, and present error estimates of the method. It follows that by pre-computing the associated harmonic coordinates, it is possible to numerically homogenize the wave equation without assumptions of scale separation or ergodicity.
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