A Compact High Order Space-Time Method for Conservation Laws (Communications in Computational Physics, 2018)
This paper presents a novel high-order space-time method for hyperbolicconservation laws. Two important concepts, the staggered space-time mesh of thespace-time conservation element/solution element (CE/SE) method and the local discontinuousbasis functions of the space-time discontinuous Galerkin (DG) finite elementmethod, are the two key ingredients of the new scheme. The staggered space-timemesh is constructed using the cell-vertex structure of the underlying spatial mesh.The universal definitions of CEs and SEs are independent of the underlying spatialmesh and thus suitable for arbitrarily unstructured meshes. The solution within eachphysical time step is updated alternately at the cell level and the vertex level. Forthis solution updating strategy and the DG ingredient, the new scheme here is termedas the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracyis achieved by employing high-order Taylor polynomials as the basis functionsinside each SE. The present DG-CVS exhibits many advantageous features such asRiemann-solver-free, high-order accuracy, point-implicitness, compactness, and easeof handling boundary conditions. Several numerical tests including the scalar advectionequations and compressible Euler equations will demonstrate the performance ofthe new method.