How you can build your own numerical discretization

This section will explain how you can build a numerical discretization of the Euler equations, in a method-of-lines spirit, that is discretizing separately the equations in time and space.

The key points are:

• choose a high-order reconstruction scheme6.1: ENO6.5, or MUSCL-TVD6.2;
• choose the limiting function in the case of MUSCL: minmod30, Van Albada32, superbee31;
• choose the variables on which the reconstruction and the limiting is performed: conservative or characteristic;
• choose the Riemann solver7 for cell-interface fluxes estimation: the exact iterative solver7.1, the Roe's approximate solver7.2, the Harten-Lax-Van Leer-Einfeldt solver7.3;

The MUSCL-TVD interpolation formula advised by van Leer6.3 provide a one parameter family of second-order scheme and one third-order scheme. You can directly select the second-order Fully Upwind scheme or the third-order one called Upwind Biased. The time-marching is achieved by applying a order two-stage Runge Kutta method, Shu, 1987.

Coupling a TVD or ENO interpolation-reconstruction technique with an appropriate Riemann solver, as said above, yields a high-order accurate scheme which numerically captures discontinuities in a sharp way avoiding unrealistic oscillations.

A first-order, upwind scheme has good properties of monotonicity, so you can use it without limiters in the discretization, but it suffers from severe inaccuracies because of the high level of truncation error.

A high-order, accurate scheme generally gives discretizations with a low-level of numerical dissipation, but steep gradients may produce oscillating behaviour due to the numerical dispersion of the scheme. Higher-order upwind schemes have been shown to optimize the distribution of the artificial dissipation reducing it in the smooth regions and to minimize the numerical dispersion.