Are Adaptive Galerkin Schemes Dissipative?

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].