Galerkin Method For Convection-diffusion Equations Research Articles

Maximum-principle-preserving third-order local discontinuous Galerkin method for convection-diffusion equations on overlapping meshes

Local discontinuous Galerkin (LDG) methods are popular for convection-diffusion equations. In LDG methods, we introduce an auxiliary variable p to represent the derivative of the primary variable u, and solve them on the same mesh. It is well known that the maximum-principle-preserving (MPP) LDG method is only available up to second-order accuracy. Recently, we introduced a new algorithm, and solve u and p on different meshes, and obtained stability and optimal error estimates. In this paper, we will continue this approach and construct MPP third-order LDG methods for convection-diffusion equations on overlapping meshes. The new algorithm is more flexible and does not increase any computational cost. Numerical evidence will be given to demonstrate the accuracy and good performance of the third-order MPP LDG method.

Spacetime discontinuous Galerkin methods for convection-diffusion equations

We have developed two new methods for solving convection-diffusion systems, with particular focus on the compressible Navier-Stokes equations. Our methods are extensions of a spacetime discontinuous Galerkin method for solving systems of hyperbolic conservation laws [3]. Following the original scheme, we use entropy variables as degrees of freedom and entropy stable numerical fluxes for the nonlinear convection term. We examine two different approaches for incorporating the diffusion term: the interior penalty method and the local discontinuous Galerkin approach. For both extensions, we can show an entropy stability result for convection-diffusion systems. Although our schemes are designed for systems, we focus on scalar convectiondiffusion equations in this contribution. This allows us to highlight our main ideas behind the stability proofs, which are the same for scalar equations and systems, in a simplified setting.

Optimal error estimates of the direct discontinuous Galerkin method for convection-diffusion equations

Abstract. In this paper, we present the optimal L2-error estimate ofO(hk+1) for polynomial elements of degree k of the semidiscrete direct discontinuous Galerkin method for convection-diffusion equations. The main technical difficulty lies in the control of the inter-element jump terms which arise because of the convection and the discontinuous nature of numerical solutions. The main idea is to use some global projections satisfying interface conditions dictated by the choice of numerical fluxes so that trouble terms at the cell interfaces are eliminated or controlled. In multi-dimensional case, the orders of k + 1 hinge on a superconvergence estimate when tensor product polynomials of degree k are used on Cartesian grids. A collection of projection errors in both oneand multi-dimensional cases is established.

High Order Maximum-Principle-Preserving Discontinuous Galerkin Method for Convection-Diffusion Equations

In this paper, we propose to apply the parametrized maximum-principle-preserving (MPP) flux limiter in [T. Xiong, J.-M. Qiu, and Z. Xu, J. Comput. Phys., 252 (2013), pp. 310--331] to the discontinuous Galerkin (DG) method for solving the convection-diffusion equations. The feasibility of applying the proposed MPP flux limiters is based on the fact that cell averages of the DG solutions are updated in a conservative fashion (by using flux difference). Compared with the earlier approach in preserving the MPP property of DG solutions for convection-diffusion problems [Y. Zhang, X. Zhang, and C.-W. Shu, J. Comput. Phys., 234 (2012), pp. 295--316], we avoid the difficulty of rewriting cell averages as a convex combination of point values in the presence of diffusion terms for DG schemes of higher than second order accuracy. As a result, our proposed parametrized MPP flux limiter can be applied to DG methods of arbitrary high order. We theoretically prove that up to the third order accuracy can be preserved for scalar one-dimensional problems. Numerical evidence is presented to show that the proposed MPP flux limiter method does not adversely affect the desired high order accuracy, nor does it require restrictive time steps. Numerical experiments including those for incompressible flow examples demonstrate the high order accuracy preserving, the MPP performance, and the robustness of the proposed method.

Analysis of variable-degree HDG methods for Convection-Diffusion equations. Part II: Semimatching nonconforming meshes

In this paper, we provide a projection-based analysis of the hversion of the hybridizable discontinuous Galerkin methods for convectiondiffusion equations on semimatching nonconforming meshes made of simplexes; the degrees of the piecewise polynomials are allowed to vary from element to element. We show that, for approximations of degree k on all elements, the order of convergence of the error in the diffusive flux is k+ 1 and that that of a projection of the error in the scalar unknown is 1 for k = 0 and k + 2 for k > 0. We also show that, for the variable-degree case, the projection of the error in the scalar variable is h-times the projection of the error in the vector variable, provided a simple condition is satisfied for the choice of the degree of the approximation on the elements with hanging nodes. These results hold for any (bounded) irregularity index of the nonconformity of the mesh. Moreover, our analysis can be extended to hypercubes.

Implicit characteristic Galerkin method for convection-diffusion equations

This paper presents a characteristic Galerkin finite element method with an implicit algorithm for solving multidimensional, time-dependent convection–diffusion equations. The method is formulated on the basis of the combination of both the precise and the implicit numerical integration procedures aiming to reference particles. The precise integration procedure with a 2N algorithm is taken as a tool to determine the material (Lagrangian) derivative of the convective function in the operator splitting procedure. The stability analysis of the algorithm and numerical results illustrate good performance of the present method in stability and accuracy. Copyright © 2000 John Wiley & Sons, Ltd.

Characteristic Galerkin method for convection-diffusion equations and implicit algorithm using precise integration

This paper presents a finite element procedure for solving transient, multidimensional convection-diffusion equations. The procedure is based on the characteristic Galerkin method with an implicit algorithm using precise integration method. With the operator splitting procedure, the precise integration method is introduced to determine the material derivative in the convection-diffusion equation, consequently, the physical quantities of material points. An implicit algorithm with a combination of both the precise and the traditional numerical integration procedures in time domain in the Lagrange coordinates for the characteristic Galerkin method is formulated. The stability analysis of the algorithm shows that the unconditional stability of present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results validate the presented method in solving convection-diffusion equations. As compared with SUPG method and explicit characteristic Galerkin method, the present method gives the results with higher accuracy and better stability.