The MMOC and MMOCAA schemes for the finite volume element method of convection-diffusion problems

We present and analyze the modified method of characteristics (MMOC) and the modified method of characteristics with adjusted advection (MMOCAA) for the finite volume element (FVE) method of convection-diffusion problems. These two schemes maintain the advantages of both the MMOC and the FVE method. And the MMOCAA scheme discussed herein conserves the conservation law globally at a minor additional computational cost. Optimal-order error estimates in the H¹-norm are proved for these schemes. A numerical example is presented to confirm the estimates.     

Analysis of a cell-vertex finite volume method for convection-diffusion problems

A cell-vertex finite volume approximation of elliptic convection-dominated diffusion equations is considered in two dimensions. The scheme is shown to be stable and second-order convergent in a mesh-dependent -norm.

     

An adaptive mesh method for 1D hyperbolic conservation laws

An adaptive method is developed for solving one-dimensional systems of hyperbolic conservation laws, which combines the rezoning approach with the finite volume weighted essentially non-oscillatory (WENO) scheme. An a posteriori error estimate, used to equidistribute the mesh, is obtained from the differences between respective numerical solutions of 5th-order WENO (WENO5) and 3rd-order ENO (ENO3) schemes. The number of grids can be adaptively readjusted based on the solution structure. For higher efficiency, mesh readjustment is performed every few time steps rather than every time step. In addition, a high order conservative interpolation is used to compute the physical solutions on the new mesh from old mesh based on the finite volume ENO reconstruction. Extensive examples suggest that this adaptive method exhibits more accurate resolution of discontinuities for a similar level of computational time comparing with that on a uniform mesh.     

A new stabilized finite volume method for the stationary Stokes equations

In this paper we develop and study a new stabilized finite volume method for the two-dimensional Stokes equations. This method is based on a local Gauss integration technique and the conforming elements of the lowest-equal order pair (i.e., the P ₁–P ₁ pair). After a relationship between this method and a stabilized finite element method is established, an error estimate of optimal order in the H ¹-norm for velocity and an estimate in the L ²-norm for pressure are obtained. An optimal error estimate in the L ²-norm for the velocity is derived under an additional assumption on the body force. This work is supported in part by the NSF of China 10701001 and by the US National Science Foundation grant DMS-0609995 and CMG Chair Funds in Reservoir Simulation.     

Semi-implicit finite volume level set method for advective motion of interfaces in normal direction

In this paper a semi-implicit finite volume method is proposed to solve the applications with moving interfaces using the approach of level set methods. The level set advection equation with a given speed in normal direction is solved by this method. Moreover, the scheme is used for the numerical solution of eikonal equation to compute the signed distance function and for the linear advection equation to compute the so-called extension speed [1]. In both equations an extrapolation near the interface is used in our method to treat Dirichlet boundary conditions on implicitly given interfaces. No restrictive CFL stability condition is required by the semi-implicit method that is very convenient especially when using the extrapolation approach. In summary, we can apply the method for the numerical solution of level set advection equation with the initial condition given by the signed distance function and with the advection velocity in normal direction given by the extension speed. Several advantages of the proposed approach can be shown for chosen examples and application. The advected numerical level set function approximates well the property of remaining the signed distance function during whole simulation time. Sufficiently accurate numerical results can be obtained even with the time steps violating the CFL stability condition.     

Two-grid methods for characteristic finite volume element solution of semilinear convection-diffusion equations

Two-grid methods for characteristic finite volume element solutions are presented for a kind of semilinear convection-dominated diffusion equations. The methods are based on the method of characteristics, two-grid method and the finite volume element method. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid size H). And the fine-grid solution (with grid size h) can be obtained by a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h^1/3).     

A stable noniterative Prediction/Correction domain decomposition method for hyperbolic problems

In this paper, a noniterative domain decomposition algorithm for solving hyperbolic partial differential equations is presented. The algorithm includes a prediction to estimate the values at the interface and these values are corrected by a scheme that improves the accuracy. It has been shown that the algorithm is unconditionally stable. Efficiency is analyzed in terms of speedup and operation ratio. Numerical experiments illustrate that the method is stable and efficient.     

A fourth-order compact ADI method for solving two-dimensional unsteady convection–diffusion problems

In this article, an exponential high-order compact (EHOC) alternating direction implicit (ADI) method, in which the Crank–Nicolson scheme is used for the time discretization and an exponential fourth-order compact difference formula for the steady-state 1D convection–diffusion problem is used for the spatial discretization, is presented for the solution of the unsteady 2D convection–diffusion problems. The method is temporally second-order accurate and spatially fourth order accurate, which requires only a regular five-point 2D stencil similar to that in the standard second-order methods. The resulting EHOC ADI scheme in each ADI solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal Gaussian decomposition and may also be solved by application of the one-dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. The unconditionally stable character of the method was verified by means of the discrete Fourier (or von Neumann) analysis. Numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order ADI method of Karaa and Zhang and the spatial third-order compact scheme of Note and Tan.     

Error estimate for the approximation of nonlinear conservation laws on bounded domains by the finite volume method

In this paper we derive a priori and a posteriori error estimates for cell centered finite volume approximations of nonlinear conservation laws on polygonal bounded domains. Numerical experiments show the applicability of the a posteriori result for the derivation of local adaptive solution strategies.

     

A symmetric characteristic finite volume element scheme for nonlinear convection-diffusion problems

In this paper, we implement alternating direction strategy and construct a symmetric FVE scheme for nonlinear convection-diffusion problems. Comparing to general FVE methods, our method has two advantages. First, the coefficient matrices of the discrete schemes will be symmetric even for nonlinear problems. Second, since the solution of the algebraic equations at each time step can be inverted into the solution of several one-dimensional problems, the amount of computation work is smaller. We prove the optimal H1-norm error estimates of order O（△t2 ＋ h） and present some numerical examples at the end of the paper.     

Convergence analysis of an upwind finite volume element method with Crouzeix-Raviart element for non-selfadjoint and indefinite problems

In this paper we construct an upwind finite volume element scheme based on the Crouzeix-Raviart nonconforming element for non-selfadjoint elliptic problems. These problems often appear in dealing with flow in porous media. We establish the optimal order H ¹-norm error estimate. We also give the uniform convergence under minimal elliptic regularity assumption      

An exponentially fitted special second-order finite difference method for solving singular perturbation problems

An exponentially fitted special second-order finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point. A fitting factor is introduced in a tri-diagonal finite difference scheme and is obtained from the theory of singular perturbations. Thomas Algorithm is used to solve the system and its stability is investigated. To demonstrate the applicability of the method, we have solved several linear and non-linear problems. From the results, it is observed that the present method approximates the exact solution very well.     

Unstructured finite volume method for matrix free explicit solution of stress-strain fields in two dimensional problems with curved boundaries in equilibrium condition

In this paper, a plane stress structural solver which uses a matrix free unstructured finite volume method based on Galerkin approach is introduced for solution of weak form of two dimensional Cauchy equations on linear triangular element meshes. The developed shape function free Galerkin finite volume structural solver explicitly computes stresses and displacements in cartesian coordinate directions for the two dimensional solid mechanic problems in equilibrium condition. The accuracy of the introduced algorithm is assessed by comparison of computed results of two plane-stress cases with curved boundaries under uniformly distributed loads with available analytical solutions. The results of the introduced method are presented in terms of stress and strain contours and its effective parameters on convergence behaviour to equilibrium condition are assessed.     

Spatial and spectral superconvergence of discontinuous Galerkin method for hyperbolic problems

In this paper, we analyze the spatial and spectral superconvergence properties of one-dimensional hyperbolic conservation law by a discontinuous Galerkin (DG) method. The analyses combine classical mathematical arguments with MATLAB experiments. Some properties of the DG schemes are discovered using discrete Fourier analyses: superconvergence of the numerical wave numbers, Radau structure of the X spatial error.     

On the coupling of finite volume and discontinuous Galerkin method for elliptic problems

The coupling of cell-centered finite volume method with primal discontinuous Galerkin method is introduced in this paper for elliptic problems. Convergence of the method with respect to the mesh size is proved. Numerical examples confirm the theoretical rates of convergence. Advantages of the coupled scheme are shown for problems with discontinuous coefficients or anisotropic diffusion matrix.     

An unstructured finite volume time domain method for structural dynamics

An unstructured finite volume time domain method (UFVTDM) is proposed to simulate stress wave propagation, in which the original variables of displacement and stress are solved based on the dynamic equilibrium equations. An Euler explicit and unstructured finite volume method is used for time dependent and spacial terms respectively. The displacements are stored on the cell vertex and a vertex based finite volume method is formed with that integral surface and the stresses are as assumed to be uniform in the cell. The present UFVTDM has several features. (1) The governing equations are discretized with the finite volume method which naturally follows conservation laws. (2) It can handle complex engineering problem. (3) This method is also able to analyze the natural characteristics and the numerical experiment shows that it is very efficient. Several cases are used to show the capability of the algorithm.     

Finite volume method for solving a one-dimensional parabolic inverse problem

In this paper, finite volume method is used to solve a one-dimensional parabolic inverse problem with source term and Neumann boundary conditions for the first time. Some advantages of this approach are developing difference schemes and maintaining certain properties of the physics of the problems, especially for the treatment of the source term and the unknown boundary conditions. Numerical results show that our method is more effective.     

Augmented high order finite volume element method for elliptic PDEs in non-smooth domains: Convergence study

The accuracy of a finite element numerical approximation of the solution of a partial differential equation can be spoiled significantly by singularities. This phenomenon is especially critical for high order methods. In this paper, we show that, if the PDE is linear and the singular basis functions are homogeneous solutions of the PDE, the augmentation of the trial function space for the Finite Volume Element Method (FVEM) can be done significantly simpler than for the Finite Element Method. When the trial function space is augmented for the FVEM, all the entries in the matrix originating from the singular basis functions in the discrete form of the PDE are zero, and the singular basis functions only appear in the boundary conditions. That is to say, there is no need to integrate the singular basis functions over the elements and the sparsity of the matrix is preserved without special care. FVEM numerical convergence studies on two-dimensional triangular grids are presented using basis functions of arbitrary high order, confirming the same order of convergence for singular solutions as for smooth solutions.