High-Order Preserving Residual Distribution Schemes for Advection-Diffusion Scalar Problems on Arbitrary Grids

Abstract

This paper deals with the construction of a class of high-order accurate residual distribution schemes for advection-diffusion problems using conformal meshes. The problems considered range from pure diffusion to pure advection. The approximation of the solution is obtained using standard Lagrangian finite elements and the total residual of the problem is constructed taking into account both the advective and the diffusive terms in order to discretize with the same scheme both parts of the governing equation. To cope with the fact that the normal component of the gradient of the numerical solution is discontinuous across the faces of the elements, the gradient of the numerical solution is reconstructed at each degree of freedom of the grid and then interpolated with the same shape functions used for the solution. Linear and nonlinear schemes are constructed and their accuracy is tested with the discretization of advectiondiffusion and anisotropic diffusion problems. Abstract. This paper deals with the construction of a class of high order accurate Residual Distribution schemes for advection-diffusion problems using conformal meshes. The problems considered range from pure difusion to pure advection. The approximation of the solution is obtained using standard Lagrangian finite elements and the total residual of the problem is constructed taking into account both the advective and the diffusive terms in order to discretize with the same scheme both parts of the governing equation. To cope with the fact that the normal component of the gradients of the numerical solution is discontinuous across the faces of the elements, the gradient of the numerical solution is recovered at each degree of freedom of the grid and then interpolated with the same shape functions used for the solution. Linear and non-linear schemes are constructed and their accuracy is tested with the discretization of advection-diffusion and anisotropic diffusion problems.1. Introduction. In the last years different high order schemes have been developed to obtain an higher order (more than two) discretization of the Navier-Stokes equations. One of the most attractive scheme seems to be the discontinuous Galerkin (DG) scheme [12]. Residual Distribution (RD) schemes [23, 1, 3] represent a very interesting alternative to DG schemes. While computationally compact and probably more flexible, DG schemes suffer from the serious drawback of a very fast growth of the number of degrees of freedom (DOF) with the cell polynomial degree. In RD schemes the formulation remains local, as in DG, but the number of DOFs growths less quickly because the solution is assumed to be continuous. Another difference between RD and DG schemes is that, to date at least, the non oscillatory properties of the RD scheme in the case of discontinuous solutions are probably better understood that for their DG counterpart.RD schemes have been developed mainly for advection problems due to possibility to construct multidimensional upwind schemes which guarantees a small disc...

Linear and non-linear high order accurate residual distribution schemes for the discretization of the steady compressible Navier–Stokes equations

“…The numerical method used to discretize the system of equations follows from the scheme already proposed by the authors in [3] to discretize the scalar advection-diffusion problem.…”

“…The linear scheme proposed in this work is the extension to the integral formulation of the classical Ni's LaxWendroff scheme [23], this scheme has been also successfully used in the context of the RD schemes to discretize the advection-diffusion scalar equation, see [3]. The nodal residual, for the DOF i of the element e, can be written as follows…”

“…The basic idea to construct a non-linear scheme is to start with a first order monotone method, and to map its nodal residuals onto a set of positive and non-linear residuals [1,3]. The first step to construct a non-linear RD method is to start with a first-order and monotone scheme.…”

“…In [3] different reconstruction strategies were considered, and it was found that the so-called Super-convergent Patch Recovery method proposed by Zienkiewicz and Zhu [38,39] (SPR-ZZ) offers the possibility to reconstruct the gradients with the same order of accuracy of the solution, without spoiling the compactness of the numerical scheme. For sake of clarity, this method is briefly recalled here.…”

“…To maintain the compactness and the efficiency of the numerical scheme, the least square or the L 2 -projection methods for the gradient reconstruction are discarded in favor of a compact and very accurate technique, the so-called Super-convergent Patch Recovery method of Zienkiewicz and Zhu [38,39]. This has already been used by the authors in the context of the RD methods, for the discretization of scalar advection-diffusion problems [3] The structure of the paper is as follows. In Section 2, the multidimensional Navier-Stokes equations for laminar flows are introduced.…”