In the compressible Euler equations, da is an identity matrix, w is the vector of conservative variables, Γ represents the conservative fluxes, and f includes the volumetric forces, heat sources, and gravity terms;

Discontinuous Galerkin methods use shape functions that are continuous in the interior of each mesh element, but discontinuous across the element boundaries. A numerical flux must be defined across element boundaries, and the dependent variables are required to be continuous on faces between adjacent mesh elements. The default numerical flux is the Lax-Friedrichs flux, which uses an average of the physical fluxes together with a penalty term. The Osher-Solomon flux is also available (see Ref. 4).

The time step in explicit computations is limited by the CFL condition; see Time Explicit Integrator in the COMSOL Multiphysics Reference Manual. The maximum stable time step is proportional to the smallest mesh element size h, and inversely proportional to the maximum wave speed in the domain. On unstructured meshes with local refinement regions only a few elements are small, yet these dictate the overall time step for the whole problem. In such cases, the use of Local Time Stepping is advisable. COMSOL Multiphysics provides the Adams-Bashforth 3 (local) time stepping method.