SIAM Review, Volume 65, Issue 4, Page 1109-1134, November 2023.
Adaptive Galerkin numerical schemes integrate time-dependent partial differential equations with a finite number of basis functions, and a subset of them is selected at each time step. This subset changes over time discontinuously according to the evolution of the solution; therefore the corresponding projection operator is time-dependent and nondifferentiable, and we propose using an integral formulation in time. We analyze the existence and uniqueness of this weak form of adaptive Galerkin schemes and prove that nonsmooth projection operators can introduce energy dissipation, which is a crucial result for adaptive Galerkin schemes. To illustrate this, we study an adaptive Galerkin wavelet scheme which computes the time evolution of the inviscid Burgers equation in one dimension and of the incompressible Euler equations in two and three dimensions with a pseudospectral scheme, together with coherent vorticity simulation which uses wavelet denoising. With the help of the continuous wavelet representation we analyze the time evolution of the solution of the 1D inviscid Burgers equation: We first observe that numerical resonances appear when energy reaches the smallest resolved scale, then they spread in both space and scale until they reach energy equipartition between all basis functions, as thermal noise does. Finally we show how adaptive wavelet schemes denoise and regularize the solution of the Galerkin truncated inviscid equations, and for the inviscid Burgers case wavelet denoising even yields convergence towards the exact dissipative solution, also called entropy solution. These results motivate in particular adaptive wavelet Galerkin schemes for nonlinear hyperbolic conservation laws. This SIGEST article is a revised and extended version of the article [R. M. Pereira, N. Nguyen van yen, K. Schneider, and M. Farge, Multiscale Model. Simul., 20 (2022), pp. 1147--1166].

中文翻译：

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自适应伽辽金方案是耗散的吗？

SIAM Review，第 65 卷，第 4 期，第 1109-1134 页，2023 年 11 月。
自适应伽辽金数值方案将瞬态偏微分方程与有限数量的基函数集成，并在每个时间步选择它们的子集。该子集根据解的演变随时间不连续地变化；因此，相应的投影算子是时间相关且不可微的，我们建议使用时间积分公式。我们分析了这种弱形式的自适应伽辽金格式的存在性和唯一性，并证明非光滑投影算子可以引入能量耗散，这是自适应伽辽金格式的关键结果。为了说明这一点，我们研究了一种自适应伽辽金小波格式，该格式使用伪谱格式计算一维无粘性伯格斯方程和二维和三维不可压缩欧拉方程的时间演化，以及使用小波去噪的相干涡度模拟。借助连续小波表示，我们分析了一维无粘 Burgers 方程解的时间演化：我们首先观察到当能量达到最小解析尺度时出现数值共振，然后它们在空间和尺度上传播，直到达到能量所有基函数之间的均分，就像热噪声一样。最后，我们展示了自适应小波方案如何对 Galerkin 截断无粘方程的解进行去噪和正则化，并且对于无粘 Burgers 情况，小波去噪甚至可以收敛到精确的耗散解（也称为熵解）。这些结果特别激发了非线性双曲守恒定律的自适应小波伽辽金方案。这篇 SIGEST 文章是文章 [RM Pereira、N. Nguyen van Yen、K. Schneider 和 M. Farge，多尺度模型。Simul.，20 (2022)，第 1147--1166 页]。