Parallel Finite Volume Computation on General Meshes

The book presents in a systematic way a methodology for the development of parallel multi-physics models and its employment in geophysical and biomedical applications. The methodology includes conservative methods of discretization of PDEs on general meshes, as well as data structures and algorithms for organization of parallel simulations on general meshes. The structures and algorithms form the core of the Integrated Numerical Modeling Object-oriented Supercomputing Technologies (INMOST) platform for the development of parallel models on general meshes. For applications, we consider geophysical and biomedical challenges. The geophysical applications address radioactive contaminant propagation with subsurface waters and reservoir simulation. The biomedical application deals with clot formation in blood flow.

Cell-centered finite volume (FV) discretizations are appealing for the approximate solution of boundary value problems since they are locally conservative and applicable to general meshes, i.e., to meshes with general polyhedral cells. In this chapter, we introduce nonlinear flux discretizations which result in monotone FV schemes at the cost of scheme nonlinearity, even if it is applied to a linear partial differential equation (PDE) such as diffusion and convection-diffusion equations. Also, we give two examples of linear two-point flux vector discretization of the diffusion equation in the mixed formulation and the Navier-Stokes equations. Such flux vector discretizations are stable in spite of degrees of freedom collocated at cell centers, are applicable to systems of PDEs, and demonstrate monotone numerical solutions.

The chapter is devoted to application of the monotone FM methods in reservoir simulation. We will consider single-phase and multi-phase black-oil models, flows in fractured media, and well-driven flows. The black-oil model is the set of PDEs that describe subsurface flow during the oil and gas recovery from natural subsurface reservoirs. The numerical modeling is the primary decision-making tool for well drilling and management. Geological surveys, core analysis, ultrasound reconnaissance, and laboratory tests are mandatory steps preceding the reservoir simulation. Reservoir simulation implies multiple numerical tests with various scenarios to coin out the best strategy for management of a particular reservoir. Multiple runs of the simulator require its computational efficiency and physically correct results. Monotone FV schemes facilitate achieving these goals.

In this chapter, the hydrogeological multi-physics models and corresponding numerical methods are presented basing on the authors’ experience of GeRa hydrogeological code [87, 93] development and its applications. Flow in unsaturated conditions, reactive transport, and density-driven flow models are addressed.

The purpose of blood flow modeling and multi-physics modeling of coagulation of blood flow is to predict thromboembolism in blood vessels and heart chambers. Cardiovascular diseases are the main causes of human death in the world. Thromboembolitic complications are one of the main reasons for strokes and infarcts. A personalized model allows one to predict clot growth under different kinds of medications and to assess the risk of blood vessel occlusion leading to new healthcare practices. In this chapter, we address our FV model for coagulation of blood flow validated by laboratory experiments.

Integrated Numerical Modeling Object-oriented Supercomputing Technologies (INMOST) is an open-source, flexible, and efficient numerical modeling framework that provides to application developers all the tools required for fast development of parallel multi-physics models. The users of INMOST avoid designing and implementing their own mesh data structures. They enjoy low-level infrastructure for reading, writing, creating, manipulating, and partitioning distributed unstructured general meshes. FV discretizations of systems of PDEs may lead to systems of nonlinear algebraic equations. The nonlinear algebraic equations are solved iteratively. On each nonlinear iteration, a sparse linear system corresponding to the nonlinear residual is assembled and solved. INMOST provides software tools that cover the complete process for parallel assembly and solution of linear systems arising in the iterative solution of nonlinear systems. To simplify the solution, the user can use INMOST tool for automatic differentiation for assembly of the nonlinear residual and the corresponding Jacobian and Hessian matrices. The synergy of the monotone FV discretizations for systems of PDEs on general meshes and INMOST instruments for development of numerical models on general meshes produces a powerful tool for supercomputing simulations. This chapter describes INMOST functionality and software mechanisms for interacting with it.