Adaptive solution of initial value problems by a dynamical Galerkin scheme

TL;DR

This paper investigates dynamical Galerkin schemes with time-varying projection operators for PDEs, demonstrating that non-smooth projections induce dissipation, which is beneficial for adaptive numerical methods like wavelet-based simulations.

Contribution

It provides a theoretical analysis of the effects of non-smooth projection operators in dynamical Galerkin schemes and illustrates their impact on energy dissipation through numerical examples.

Findings

Non-smooth projections cause dissipation of energy.

Thresholding wavelet coefficients introduces dissipation.

Numerical experiments confirm dissipation effects in Burgers and Euler equations.

Abstract

We study dynamical Galerkin schemes for evolutionary partial differential equations (PDEs), where the projection operator changes over time. When selecting a subset of basis functions, the projection operator is non-differentiable in time and an integral formulation has to be used. We analyze the projected equations with respect to existence and uniqueness of the solution and prove that non-smooth projection operators introduce dissipation, a result which is crucial for adaptive discretizations of PDEs, e.g., adaptive wavelet methods. For the Burgers equation we illustrate numerically that thresholding the wavelet coefficients, and thus changing the projection space, will indeed introduce dissipation of energy. We discuss consequences for the so-called `pseudo-adaptive' simulations, where time evolution and dealiasing are done in Fourier space, whilst thresholding is carried out in wavelet space. Numerical examples are given for the inviscid Burgers equation in 1D and the incompressible Euler equations in 2D and 3D.