Numerical study on the convergence to steady state solutions of a new class of finite volume WENO schemes: triangular meshes

In this paper we continue our research on the numerical study of convergence to steady state solutions for a new class of finite volume weighted essentially non-oscillatory (WENO) schemes in [38], from tensor product meshes to triangular meshes. For the case of triangular meshes, this new class of finite volume WENO schemes was designed for time-dependent conservation laws in [37] for the third and fourth order version and in this paper for the fifth order version, with the main idea being the application of constructing one high degree polynomial on a big central stencil to get high order approximation in smooth regions and several linear polynomials defined on small stencils located centrally and in different sectorial regions which are partitioned by the barycenter and three vertices of the triangular cells to keep the essentially non-oscillatory property near discontinuities. Similar to the case of tensor product meshes in [38], by performing such spatial reconstruction procedures together with a TVD Runge-Kutta time discretization, these WENO schemes do not suffer from slight post-shock oscillations which are responsible for the residue of classical WENO schemes to hang at a truncation error level instead of converging to machine zero. The third, fourth and fifth order finite volume WENO schemes in this paper can suppress the slight post-shock oscillations and have their residues settling down to a tiny number close to machine zero in steady state simulations in our extensive numerical experiments.