Space-Time Adaptive Multiresolution Techniques for Compressible Euler Equations

This paper considers space-time adaptive techniques for finite volume schemes with explicit time discretization. The purpose is to reduce memory and to speed-up computations by a multiresolution representation of the numerical solution on adaptive grids which are introduced by suitable thresholding of its wavelet coefficients. Further speed-up is obtained by the combination of the multiresolution scheme with an adaptive strategy for time integration, which is classical for ODE simulations. It considers variable time steps, controlled by a given precision, using embedded Runge–Kutta schemes. As an alternative to the celebrated CFL condition, the aim in the application of such an time-adaptive scheme for PDE simulations is to obtain accurate and safe integrations. The efficiency of this adaptive space-time method is analyzed in applications to typical Riemann–Lax test problems for the compressible Euler equations in one and two space dimensions. The results show that the accuracy properties of the reference finite volume scheme on the finest regular grid, where the time step is determined by the CFL condition, is preserved. Nevertheless, both CPU time and memory requirements are considerably reduced, thanks to the efficient self-adaptive grid refinement and controlled time-stepping.