Adaptive Solution of Initial Value Problems by a Dynamical Galerkin Scheme
R. M. Pereira, Natacha Nguyen van yen, Kai Schneider, Marie Farge · Multiscale Modeling and Simulation · 2022
Abstract. Adaptive Galerkin methods for time-dependent partial differential equations are studied and shown to be dissipative. The adaptation implies that the subset of the selected basis function changes over time according to the evolution of the solution. The corresponding projection operator is thus time-dependent and nondifferentiable. We therefore propose to use an integral formulation in time. We analyze the existence and uniqueness of this weak form of the dynamical Galerkin scheme, and we then prove that the nonsmooth projection operator introduces energy dissipation, which is a crucial result for adaptive Galerkin methods, e.g., adaptive wavelet methods. Numerical examples for the inviscid Burgers equation in one dimension and the incompressible Euler equations in two and three spatial dimensions show that the selection of basis functions, for instance by filtering out weak wavelet coefficients from the solution, introduces energy dissipation. Moreover, for the Burgers case we can show that adaptive wavelet regularization yields convergence of the truncated Galerkin solution to the physically relevant entropy solution. These results motivate adaptive wavelet-based Galerkin schemes for nonlinear hyperbolic conservation laws.