A high-order conservative collocation scheme and its application to global shallow-water equations

Abstract

Abstract. In this paper, an efficient and conservative collocation method is proposed and used to develop a global shallow-water model. Being a nodal type high-order scheme, the present method solves the pointwise values of dependent variables as the unknowns within each control volume. The solution points are arranged as Gauss-Legendre points to achieve high-order accuracy. The time evolution equations to update the unknowns are derived under the flux reconstruction (FR) framework (Huynh, 2007). Constraint conditions used to build the spatial reconstruction for the flux function include the pointwise values of flux function at the solution points, which are computed directly from the dependent variables, as well as the numerical fluxes at the boundaries of the computational element, which are obtained as Riemann solutions between the adjacent elements. Given the reconstructed flux function, the time tendencies of the unknowns can be obtained directly from the governing equations of differential form. The resulting schemes have super convergence and rigorous numerical conservativeness.A three-point scheme of fifth-order accuracy is presented and analyzed in this paper. The proposed scheme is adopted to develop the global shallow-water model on the cubedsphere grid, where the local high-order reconstruction is very beneficial for the data communications between adjacent patches. We have used the standard benchmark tests to verify the numerical model, which reveals its great potential as a candidate formulation for developing high-performance general circulation models.

Entropy stable wall boundary conditions for the three-dimensional compressible Navier–Stokes equations

“…14 Recall that the diagonal element of any operator P are equal to the spacing between flux points. 15 This dependence on the mesh size in the normal direction to the face is similar to that of the interior penalty approach used in finite element methods. thermal condition, is proven to be bounded by physical data provided by the user.…”

“…This factor is also important because it allows to achieve the correct asymptotic order of accuracy and yields an increase in the strength of M with increased resolution. 15 Summarizing Equation ( 54), the penalty at the face point is the sum of three terms:…”

“…In computational fluid dynamics (CFD), next generation numerical algorithms for use in large eddy simulations (LES) and hybrid Reynolds-averaged Navier-Stokes (RANS)-LES simulations will undoubtedly rely on efficient high-order accurate formulations (see, for instance, [2,3,4,5,6,7,8,9,10,11,12,13,14,15]). Although high order techniques are well suited for smooth solutions, numerical instabilities may occur if the flow contains discontinuities or under-resolved physical features.…”

High-order numerical solutions to the shallow-water equations on the rotated cubed-sphere grid

“…It has been shown that the DFR method results in a scheme equivalent to the weak form of the nodal discontinuous Galerkin method in one dimension and on multidimensional elements with tensor product bases [1,3]. It should be noted that these ideas are not entirely new in geophysical applications since the 3rd order scheme presented in [31] may also be considered as a special case of the DFR method. In § 4.1, an overview of the basic properties of the method in one dimension is presented and its extensions to two dimensions and curved geometry is discussed in § 4.2.…”