On increasing the stability of the combined scheme of the discontinuous Galerkin method

A special modification of the combined scheme of the discontinuous Galerkin method is proposed, which increases the stability of this scheme when calculating discontinuous solutions with shock waves. This modification is associated with the addition of an artificial viscosity of the fourth order of divergence to the basic scheme included in this combined scheme. Test calculations are presented. They demonstrate the advantages of the new combined scheme in comparison with the standard monotonized versions of the discontinuous Galerkin method.