Space‐time residual distribution schemes for hyperbolic conservation laws on unstructured linear finite elements

Abstract

Multidimensional upwind residual distribution schemes are extended to the context of continuous linear space-time ÿnite elements for the time accurate solution of scalar and hyperbolic systems of conservation laws. The formulation leads to a consistent discretization of the space-time domain, thus retaining the properties of the underlying basic schemes both in space and time. We propose a particular space -time mesh conÿguration containing two layers of elements and three levels of nodes in time. This construction leads to unconditionally stable implicit time stepping while retaining second-order spatial and temporal accuracy in smooth ows and monotone solution across steep gradients. The presented schemes have a strong potential in the ÿeld of moving grids, since they allow a dynamic change of the space-time mesh geometry. Numerical results demonstrate the robustness, accuracy and monotonicity of the method.

Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes

“…with Dt E the value of the time-step guaranteeing [31,8,7] (see also Section 3) X . Both in steady and unsteady computations, the nonlinear scheme has been implemented as described in Section 3.5, settingñ ¼ũ (the velocity vector) for the wave decomposition.…”

“…Alternatively, one can use the formulation of [31]. In this case, we refer to the space-time ST-N scheme as the one defined by the local nodal residuals (see Eq.…”

“…Under a proper time-step restriction [8,31] (see also Section 4), the eigenvalues of all the C n j matrices are negative. A conservative and space-time MU scheme is then obtained as…”

“…In [8,31] it is shown that the C j matrices are true generalizations to space-time prismatic elements of the K j Jacobians. Under a proper time-step restriction [8,31] (see also Section 4), the eigenvalues of all the C n j matrices are negative.…”

High order residual distribution conservative finite difference WENO schemes for steady state problems on non-smooth meshes

“…For a hybrid of different meshes, quadrilateral meshes have also been considered [31], and a recent work by Abgrall and Marpeau [4] gives a formal construction of second order schemes on quadrilateral meshes. Besides steady state problems, attention has also been paid recently to design RD schemes for unsteady problems [11,6].…”