Finite Volume Element Schemes Research Articles

A symmetric characteristic FVE method with second order accuracy for nonlinear convection diffusion problems

The finite volume element (FVE) methods used currently are essentially low order and unsymmetric. In this paper, by biquadratic elements and multistep methods, we construct a second order FVE scheme for nonlinear convection diffusion problem on nonuniform rectangular meshes. To overcome the numerical oscillation, we discretize the problem along its characteristic direction. The choice of alternating direction strategy is critical in this paper, which guarantees the high efficiency and symmetry of the discrete scheme. Optimal order error estimates in H 1 -norm are derived and a numerical example is given at the end to confirm the usefulness of the method.

High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms

Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source term. In our earlier work [J. Comput. Phys. 208 (2005) 206–227; J. Sci. Comput., accepted], we designed a well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, which at the same time maintains genuine high order accuracy for general solutions, to a class of hyperbolic systems with separable source terms including the shallow water equations, the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. In this paper, we generalize high order finite volume WENO schemes and Runge–Kutta discontinuous Galerkin (RKDG) finite element methods to the same class of hyperbolic systems to maintain a well-balanced property. Finite volume and discontinuous Galerkin finite element schemes are more flexible than finite difference schemes to treat complicated geometry and adaptivity. However, because of a different computational framework, the maintenance of the well-balanced property requires different technical approaches. After the description of our well-balanced high order finite volume WENO and RKDG schemes, we perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions.

A mixed finite volume element method based on rectangular mesh for biharmonic equations

This paper presents a mixed finite volume element scheme based on rectangular partition for solving biharmonic equations. It also gives a kind of adaptive Uzawa iteration method for the scheme. It is rigorously proved that the scheme has first-order accuracy in H 1 semi-norm and L 2 norm according to the characteristics of the scheme. Finally, two numerical examples illustrate the effectiveness of the method.

Some aspects of adaptive grid technology related to boundary and interior layers

We consider the use of adaptive mesh strategies for solution of problems exhibiting boundary and interior layer solutions. As the presence of these layer structures suggests, reliable and accurate solution of this class of problems using finite difference, finite volume or finite element schemes requires grading the mesh into the layers and due attention to the associated algorithms. When the nature and structure of the layer is known, mesh grading can be achieved during the grid generation by specifying an appropriate grading function. However, in many applications the location and nature of the layer behavior is not known in advance. Consequently, adaptive mesh techniques that employ feedback from intermediate grid solutions are an appealing approach. In this paper, we provide a brief overview of the main adaptive grid strategies in the context of problems with layers. Associated error indicators that guide the refinement feedback control/grid optimization process are also covered and there is a brief commentary on the supporting data structure requirements. Some current issues concerning the use of stabilization in conjunction with adaptive mesh refinement (AMR), the question of “pollution effects” in computation of local error indicators, the influence of nonlinearities and the design of meshes for targeted optimization of specific quantities are considered. The application of AMR for layer problems is illustrated by means of case studies from semiconductor device transport (drift diffusion), nonlinear reaction–diffusion, layers due to surface capillary effects, and shockwaves in compressible gas dynamics.

On discontinuous Galerkin methods for nonlinear convection-diffusion problems and compressible flow

The paper is concerned with the discontinuous Galerkin finite element method for the numerical solution of nonlinear conservation laws and nonlinear convection-diffusion problems with emphasis on applications to the simulation of compressible flows. We discuss two versions of this method: (a) Finite volume discontinuous Galerkin method, which is a generalization of the combined finite volume—finite element method. Its advantage is the use of only one mesh (in contrast to the combined finite volume—finite element schemes). However, it is of the first order only. (b) Pure discontinuous Galerkin finite element method of higher order combined with a technique avoiding spurious oscillations in the vicinity of shock waves.

Finite Volume Element Approximations of Nonlocal in Time One-Dimensional Flows in Porous Media

Various finite volume element schemes for parabolic integro-differential equations in 1-D are derived and studied. These types of equations arise in modeling reactive flows or material with memory effects. Our main goal is to develop a general framework for obtaining finite volume element approximations and to study the error analysis. We consider the lowest-order (linear and L-splines) finite volume elements, although higher-order volume elements can be considered as well under this framework. It is proved that finite volume element approximations are convergent with optimal order in H 1-norms, suboptimal order in the L 2-norm and super-convergent order in a discrete H 1-norm.

Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations

Numerical simulations of flow problems with moving boundaries commonly require the solution of the fluid equations on unstructured and deformable dynamic meshes. In this paper, we present a unified theory for deriving Geometric Conservation Laws (GCLs) for such problems. We consider several popular discretization methods for the spatial approximation of the flow equations including the Arbitrary Lagrangian-Eulerian (ALE) finite volume and finite element schemes, and space-time stabilized finite element formulations. We show that, except for the case of the space-time discretization method, the GCLs impose important constraints on the algorithms employed for time-integrating the semi-discrete equations governing the fluid and dynamic mesh motions. We address the impact of these constraints on the solution of coupled aeroelastic problems, and highlight the importance of the GCLs with an illustration of their effect on the computation of the transient aeroelastic response of a flat panel in transonic flow.

A finite volume convergence theory for the fast adaptive composite grid methods

The fast adaptive composite grid method (FAC) is an efficient multilevel technique for adaptive solution of partial differential equations. Most existing FAC convergence theory assumes that the discrete equations and interlevel transfers are variational, which restricts applicability primarily to Galerkin finite element discretizations of self-adjoint elliptic equations. This paper extends the variational two-level theory to the case that the so-called finite volume element scheme is used for discretization.

A streamline upwind control volume finite element method for modeling fluid flow and heat transfer problems

Galerkin and control volume finite element approximations are compared and contrasted. Streamline upwinding and pressure stabilizing Petrov-Galerkin formulations for the numerical modeling of fluid flow problems are discussed. These formulations are then adapted to a control volume finite element discretization. The resulting control volume finite element scheme permits an equal order interpolation for velocity and pressure, is readily implemented and retains the property of local conservation, normally associated with control volume schemes.

Studies in Bayesian Econometrics and Statistics, In honor of Leonard J. Savage

We construct a finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes, where the strict convexity restriction on the meshes is removed. The main contributions of this paper include three aspects. Firstly, the classical finite volume element (C-FVE) method is extended to severely distorted quadrilateral meshes even with concave cells by virtue of a new overlapping dual partition and a new gradient approximation. In fact, the choice of this dual partition and gradient approximation is also conducive to the construction of a monotone scheme. Secondly, a new monotonicity correction is suggested, based on which we obtain a monotone finite volume element (M-FVE) method. The resulting M-FVE method still keeps the local conservation and is easy for implementation. Finally, we analyze theoretically the truncation error and the monotonicity for this scheme. Besides, the existence of a solution to this nonlinear scheme is proved by applying the Brouwer's fixed point theorem. Numerical results demonstrate that the M-FVE method has the approximate second-order accuracy and preserves well the positivity of the solution for both isotropic and anisotropic diffusion problems on severely distorted quadrilateral meshes.