Space–time adaptive multiresolution methods for hyperbolic conservation laws: Applications to compressible Euler equations

Abstract

Adaptive strategies in space and time allow considerable speed-up of finite volume schemes for conservation laws, while controlling the accuracy of the discretization. In this paper, a multiresolution technique for finite volume schemes with explicit time discretization is presented. An adaptive grid is introduced by suitable thresholding of the wavelet coefficients, which maintains the accuracy of the finite volume scheme of the regular grid. Further speed-up is obtained by local scale-dependent time stepping, i.e., on large scales larger time steps can be used without violating the stability condition of the explicit scheme. Furthermore, an estimation of the truncation error in time, using embedded Runge–Kutta type schemes, guarantees a control of the time step for a given precision. The accuracy and efficiency of the fully adaptive method is illustrated with applications for compressible Euler equations in one and two space dimensions.