Mikhail J. Shashkov

Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes

The stability and convergence properties of the mimetic finite difference method for diffusion-type problems on polyhedral meshes are analyzed. The optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved.

Natural discretizations for the divergence, gradient, and curl on logically rectangular grids☆

Abstract This is the first in series of papers creating a discrete analog of vector analysis on logically rectangular, nonorthogonal, nonsmooth grids . We introduce notations for 2-D logically rectangular grids, describe both cell-valued and nodal discretizations for scalar functions, and construct the natural discretizations of vector fields, using the vector components normal and tangential to the cell boundaries. We then define natural discrete analogs of the divergence, gradient, and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates, such as length of elements of coordinate lines, areas of elements of coordinate surfaces, and elementary volumes. We introduce the discrete volume integral of scalar functions, the discrete surface integral, and a discrete analog of the line integral and prove discrete versions of the main theorems relating these objects. These theorems include the following: the discrete analog of relationship div A → = 0 if and only if A → = curl B → ; curl A → = 0 if and only if A → = grad ϕ ; if A → = grad ϕ , then the line integral does not depend on path; and if the line integral of a vector function is equal to zero for any closed path, then this vector is the gradient of a scalar function. Last, we define the discrete operators DIV, GRAD, and CURL in terms of primitive differencing operators (based on forward and backward differences) and primitive metric operators (related to multiplications of discrete functions by length of edges, areas of surfaces, and volumes of 3-D cells). These formulations elucidate the structure of the discrete operators and are useful when investigating the relationships between operators and their adjoints.

Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes

We consider a non-linear finite volume (FV) scheme for stationary diffusion equation. We prove that the scheme is monotone, i.e. it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients. The scheme is extended to regular star-shaped polygonal meshes and isotropic heterogeneous coefficients.

Mimetic finite difference method

The mimetic finite difference (MFD) method mimics fundamental properties of mathematical and physical systems including conservation laws, symmetry and positivity of solutions, duality and self-adjointness of differential operators, and exact mathematical identities of the vector and tensor calculus. This article is the first comprehensive review of the 50-year long history of the mimetic methodology and describes in a systematic way the major mimetic ideas and their relevance to academic and real-life problems. The supporting applications include diffusion, electromagnetics, fluid flow, and Lagrangian hydrodynamics problems. The article provides enough details to build various discrete operators on unstructured polygonal and polyhedral meshes and summarizes the major convergence results for the mimetic approximations. Most of these theoretical results, which are presented here as lemmas, propositions and theorems, are either original or an extension of existing results to a more general formulation using polyhedral meshes. Finally, flexibility and extensibility of the mimetic methodology are shown by deriving higher-order approximations, enforcing discrete maximum principles for diffusion problems, and ensuring the numerical stability for saddle-point systems.

Second-order sign-preserving conservative interpolation (remapping) on general grids

An accurate conservative interpolation (remapping) algorithm is an essential component of most arbitrary Lagrangian-Eulerian (ALE) methods. In this paper we describe a local remapping algorithm for a positive scalar function. This algorithm is second-order accurate, conservative, and sign preserving. The algorithm is based on estimating the mass exchanged between cells at their common interface, and so is equally applicable to structured and unstructured grids. We construct the algorithm in a series of steps, clearly delineating the assumptions and errors made at each step. We validate our theory with a suite of numerical examples, analyzing the results from the viewpoint of accuracy and order of convergence.

Multi-material interface reconstruction on generalized polyhedral meshes

We describe multi-material (more than two materials) interface reconstruction methods for 3D meshes of generalized polyhedra. The basic information used in interface reconstruction is the volume fraction of each material in mixed cells, that is, those containing multiple materials. All methods subdivide a mixed cell into a set of pure non-overlapping sub-cells, each containing just one material that have the reference volume fraction. We describe three methods. The first two methods represent an extension of standard piece-wise linear interface construction (PLIC) methods to 3D and use information only about volume fractions. The first method is first-order accurate and is based on the discrete gradient of the volume fraction as an estimate of the normal to the interface. The second method is planarity-preserving (second-order accurate) and is an extension to 3D of the least squares volume-of-fluid interface reconstruction algorithm (LVIRA, see E. Puckett, A volume-of-fluid interface tracking algorithm with applications to computing shock wave refraction, in: H. Dwyer (Ed.), Proceedings of the Fourth International Symposium on Computational Fluid Dynamics, 1991, pp. 933-938; J.E. Pilliod, E.G. Puckett, Second-order accurate volume-of-fluid algorithms for tracking material interfaces, Journal of Computational Physics 199 (2004) 465-502] for the 2D case). The third method is an extension to 3D of the so-called moment-of-fluid (MoF) method V. Dyadechko, M. Shashkov, Moment-of-fluid interface reconstruction, Tech. Rep. LA-UR-05-7571, Los Alamos National Laboratory, 2005. Also available as http://cnls.lanl.gov/~shashkov/; V. Dyadechko, M. Shashkov, Multi-material interface reconstruction from the moment data, Tech. Rep. LA-UR-06-5846, Los Alamos National Laboratory, 2006. Also available as http://cnls.lanl.gov/~shashkov/]. The MoF method is also second-order accurate. This method uses information not only about volume fractions but also about the position of the centroids of each material. In contrast to standard PLIC methods, the MoF method uses only information from the cell where reconstruction is performed; no information from neighboring cells is needed. Also, the MoF method provides automatic ordering of the materials during interface reconstruction. Optimal ordering is based on comparing the positions of the reference centroids and actual centroids of the reconstructed pure sub-cells. The performance of the methods is demonstrated with numerical examples.

Reconstruction of multi-material interfaces from moment data

The moment-of-fluid (MoF) method is an extension of popular volume-of-fluid (VoF) technique for tracking material interface in multi-material fluid flows. VoF methods track the cell-wise material volumes and use these data for reconstructing the interfaces in mixed cell. The MoF method goes one step further and, in additional to the volumes, keeps track of the cell-wise material centroids; this approach provides sufficiently more information for the interface reconstruction algorithm. The MoF algorithm reconstructs interfaces in volume-conservative manner, by minimizing the defect of the 1st moment in each mixed cell. In case of two materials, this strategy allows to construct the linear interface in a mixed cell using no material volume data from the neighboring cells. Compared to the VoF interface reconstruction techniques, the MoF algorithm shows higher accuracy and better resolution, allows uniform processing of internal and boundary cells. In this paper we show how the same governing principle (minimization of the 1st-moment defect) can be used to reconstruct the interfaces in case of multiple materials.

Mimetic Finite Difference Methods for Diffusion Equations

This paper reviews and extends the theory and application of mimetic finite difference methods for the solution of diffusion problems in strongly heterogeneous anisotropic materials. These difference operators satisfy the fundamental identities, conservation laws and theorems of vector and tensor calculus on nonorthogonal, nonsmooth, structured and unstructured computational grids. We provide explicit approximations for equations in two dimensions with discontinuous anisotropic diffusion tensors. We mention the similarities and differences between the new methods and mixed finite element or hybrid mixed finite element methods.

The Orthogonal Decomposition Theorems for Mimetic Finite Difference Methods

Accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus provide reliable finite difference methods for approximating the solutions to a wide class of partial differential equations. These methods mimic many fundamental properties of the underlying physical problem including conservation laws, symmetries in the solution, and the nondivergence of particular vector fields (i.e., they are divergence free) and should satisfy a discrete version of the orthogonal decomposition theorem. This theorem plays a fundamental role in the theory of generalized solutions and in the numerical solution of physical models, including the Navier--Stokes equations and in electrodynamics. We are deriving mimetic finite difference approximations of the divergence, gradient, and curl that satisfy discrete analogs of the integral identities satisfied by the differential operators. We first define the natural discrete divergence, gradient, and curl operators based on coordinate invariant definitions, such as Gausss theorem, for the divergence. Next we use the formal adjoints of these natural operators to derive compatible divergence, gradient, and curl operators with complementary domains and ranges of values. In this paper we prove that these operators satisfy discrete analogs of the orthogonal decomposition theorem and demonstrate how a discrete vector can be decomposed into two orthogonal vectors in a unique way, satisfying a discrete analog of the formula

Local flux mimetic finite difference methods

\vec{A} = \ggrad \, \varphi + \curl \, \vec{B}