Yuri V. Vassilevski

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.

Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes

We developed a new monotone finite volume method for diffusion equations. The second-order linear methods, such as the multipoint flux approximation, mixed finite element and mimetic finite difference methods, are not monotone on strongly anisotropic meshes or for diffusion problems with strongly anisotropic coefficients. The finite volume (FV) method with linear two-point flux approximation is monotone but not even first-order accurate in these cases. The developed monotone method is based on a nonlinear two-point flux approximation. It does not require any interpolation scheme and thus differs from other nonlinear finite volume methods based on a two-point flux approximation. The second-order convergence rate is verified with numerical experiments.

Iterative Solution Methods for Modeling Multiphase Flow in Porous Media Fully Implicitly

We discuss several fully implicit techniques for solving the nonlinear algebraic system arising in an expanded mixed finite element or cell-centered finite difference discretization of two- and three-phase porous media flow. Every outer nonlinear Newton iteration requires solution of a nonsymmetric Jacobian linear system. Two major types of preconditioners, supercoarsening multigrid (SCMG) and two-stage, are developed for the GMRES iteration applied to the solution of the Jacobian system. The SCMG reduces the three-dimensional system to two dimensions using a vertical aggregation followed by a two-dimensional multigrid. The two-stage preconditioners are based on decoupling the system into a pressure and concentration equations. Several pressure preconditioners of different types are described. Extensive numerical results are presented using the integrated parallel reservoir simulator (IPARS) and indicate that these methods have low arithmetical complexity per iteration and good convergence rates.

Pressure Schur Complement Preconditioners for the Discrete Oseen Problem

We consider several preconditioners for the pressure Schur complement of the discrete steady Oseen problem. Two of the preconditioners are well known from the literature and the other is new. Supplemented with an appropriate approximate solve for an auxiliary velocity subproblem, these approaches give rise to a family of the block preconditioners for the matrix of the discrete Oseen system. In the paper we critically review possible advantages and difficulties of using various Schur complement preconditioners. We recall existing eigenvalue bounds for the preconditioned Schur complement and prove such for the newly proposed preconditioner. These bounds hold both for LBB stable and stabilized finite elements. Results of numerical experiments for several model two-dimensional and three-dimensional problems are presented. In the experiments we use LBB stable finite element methods on uniform triangular and tetrahedral meshes. One particular conclusion is that in spite of essential improvement in comparison with “simple” scaled mass-matrix preconditioners in certain cases, none of the considered approaches provides satisfactory convergence rates in the case of small viscosity coefficients and a sufficiently complex (e.g., circulating) advection vector field.

A monotone finite volume method for advection-diffusion equations on unstructured polygonal meshes

We present a new second-order accurate monotone finite volume (FV) method for the steady-state advection-diffusion equation. The method uses a nonlinear approximation for both diffusive and advective fluxes and guarantees solution non-negativity. The interpolation-free approximation of the diffusive flux uses the nonlinear two-point stencil proposed in Lipnikov [23]. Approximation of the advective flux is based on the second-order upwind method with a specially designed minimal nonlinear correction. The second-order convergence rate and monotonicity are verified with numerical experiments.

Decoupling preconditioners in the implicit parallel accurate reservoir simulator (IPARS)

This paper presents an overview of two-stage decoupling preconditioning techniques employed in the implicit parallel accurate reservoir simulator (IPARS) computational framework for modelling multi-component multi-phase flow in porous media. The underlying discretization method is implicit Euler in time and mixed finite elements or cell-centred finite differences in space. IPARS permits rigorous, physically representative coupling of different physical and numerical flow models in different parts of the domain and accounts for structural discontinuities; the framework currently includes eight physical models. For simplicity of exposition, we have restricted our discussion to a two-phase oil–water model and a three-phase black oil model. Our decoupling approach involves extracting a pressure equation from the fully coupled linearized system thus allowing for a more accurate preconditioning of a discrete elliptic problem of lower dimension. Copyright

A parallel solver for unsteady incompressible 3D Navier—Stokes equations

Abstract We describe some algorithms and software components that allow us to solve on parallel computers classical test cases for unsteady incompressible 3D Navier–Stokes equations. Our main focus is the design of robust and efficient parallel solvers for systems with singularly perturbed convection–reaction–diffusion and Laplace operators, which are important constituents of the Navier–Stokes solvers. The performance of the solvers on two parallel computers is examined.

A monotone nonlinear finite volume method for diffusion equations and multiphase flows

We present a new nonlinear monotone finite volume method for diffusion equation and its application to two-phase flow model. We consider full anisotropic discontinuous diffusion or permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which provides the conventional seven-point stencil for the discrete diffusion operator on cubic meshes. We show that the quality of the discrete flux in a reservoir simulator has great effect on the front behavior and the water breakthrough time. We compare two two-point flux approximations (TPFA), the proposed nonlinear TPFA and the conventional linear TPFA, and multipoint flux approximation (MPFA). The new nonlinear scheme has a number of important advantages over the traditional linear discretizations. Compared to the linear TPFA, the nonlinear TPFA demonstrates low sensitivity to grid distortions and provides appropriate approximation in case of full anisotropic permeability tensor. For nonorthogonal grids or full anisotropic permeability tensors, the conventional linear TPFA provides no approximation, while the nonlinear flux is still first-order accurate. The computational work for the new method is higher than the one for the conventional TPFA, yet it is rather competitive. Compared to MPFA, the new scheme provides sparser algebraic systems and thus is less computational expensive. Moreover, it is monotone which means that the discrete solution preserves the nonnegativity of the differential solution.

Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations

Contemporary time stepping schemes applied to the solution of unsteady nonlinear fluid flow problems are considered. The iterative solution of arising series of linear and nonlinear systems and the choice of the initial guess are addressed. The computation of a better initial guess for two iterative linear system solvers (GCR and GMRES) is based on the history of the evolution problem solving. For implicitly discretized nonlinear evolution problems, a reduced model technique is developed for computing a better initial guess for the inexact Newton method. The computational effect of the chosen initial guess is compared with that of the standard (physically motivated) initial guess.

ILU Preconditioners for Nonsymmetric Saddle-Point Matrices with Application to the Incompressible Navier--Stokes Equations

Motivated by the numerical solution of the linearized incompressible Navier--Stokes equations, we study threshold incomplete LU factorizations for nonsymmetric saddle-point matrices. The resulting preconditioners are used to accelerate the convergence of a Krylov subspace method applied to finite element discretizations of fluid dynamics problems in three space dimensions. The paper presents and examines an extension for nonsymmetric matrices of the Tismenetsky--Kaporin incomplete factorization. It is shown that in numerically challenging cases of higher Reynolds number flows one benefits from using this two-parameter modification of a standard threshold ILU preconditioner. The performance of the ILU preconditioners is studied numerically for a wide range of flow and discretization parameters, and the efficiency of the approach is shown if threshold parameters are chosen suitably. The practical utility of the method is further demonstrated for the haemodynamic problem of simulating blood flow in a right c...