Numerical Initial Value Problems in Ordinary Differential Equations

Publisher Summary This chapter focuses on finite volume methods. The finite volume method is a discretization method that is well suited for the numerical simulation of various types (for instance, elliptic, parabolic, or hyperbolic) of conservation laws; it has been extensively used in several engineering fields, such as fluid mechanics, heat and mass transfer, or petroleum engineering. Some of the important features of the finite volume method are similar to those of the finite element method: it may be used on arbitrary geometries, using structured or unstructured meshes, and it leads to robust schemes. The finite volume method is locally conservative because it is based on a “balance" approach: a local balance is written on each discretization cell that is often called “control volume;” by the divergence formula, an integral formulation of the fluxes over the boundary of the control volume is then obtained. The fluxes on the boundary are discretized with respect to the discrete unknowns.

http://materias.fi.uba.ar/7538/material/Otros/EymardEtAl%20-%20Finite%20Volume%20Methods.pdf

Finite Volume Methods

This paper provides an error analysis for the Crank–Nicolson method of time discretization applied to spatially discrete Galerkin approximations of the nonstationary Navier–Stokes equations. Second-order error estimates are proven locally in time under realistic assumptions about the regularity of the solution. For approximations of an exponentially stable solution, these local error estimates are extended uniformly in time as ${\text{t}} \to \infty $.

Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization

Introduction The importance of a quantitative understanding of the hydrological cycle increases with the ever-growing demand for water for anthropogenic needs. Numerical models are inevitable tools in this undertaking. A wide range of numerical models of different complexity have been developed for this purpose, ranging from simple, lumped parameter models to more complex, physically based models. The foundation of physically based models is the blueprint paper by Freeze and Harlan (1969), and numerous physically based models have been developed following this blueprint. HydroGeoSphere (HGS), the code discussed in this review, is one of them. The origin of HGS is the code FRAC3DVS, developed by R. Therrien at the University of Waterloo as part of his doctoral work under the supervision of E.A. Sudicky (Therrien 1992). FRAC3DVS was designed to simulate variably saturated groundwater flow and advective-dispersive solute transport in porous or discretely fractured porous media. In 2002, a two-dimensional (2D) surface water flow and transport component were implemented in FRAC3DVS and the code was renamed HydroGeoSphere. Until recently, the code was free for academic research, while commercial users paid a license fee between 3000 and 6000 US dollars depending on the number of CPU cores the code will use in a parallel computational platform. The code can be downloaded by contacting the developers through the website: http://hydrogeosphere.org/. HGS has been designed to solve simple problems (e.g., regular geometry, steady-state saturated flow etc.) as well as very complex problems (e.g., integrated flow,

/pdf/hydrogeosphere-a-fully-integrated-physically-based-4xnni8j5l0.pdf

HydroGeoSphere: A Fully Integrated, Physically Based Hydrological Model

Three-dimensional analysis of variably-saturated flow and solute transport in discretely-fractured porous media

This paper describes a fully coupled geochemical compositional Equation-of-State (EOS) compositional simulator for the simulation of CO2 storage in saline aquifers. The simulator (GEM-GHG) models the following phenomena: (1) convective and dispersive flow in porous media; (2) phase equilibrium between the oil, gas and aqueous phase; (3) chemical equilibrium for reactions between the aqueous components and (4) mineral dissolution and precipitation kinetics. For numerical robustness and stability, all equations are solved simultaneously. The simulator is applied to the simulation of typical field-scale CO2 sequestration processes, showing the migration of CO2(g) and CO2(aq), the dissociation of CO2(aq) into − HCO and its subsequent conversion into carbonate minerals. Convection of highdensity plumes of CO2-rich brine in conjunction with CO2 mineralization around the plumes is illustrated.

Modeling CO2 Storage in Aquifers with a Fully-Coupled Geochemical EOS Compositional Simulator

Quadratic convergence for cell-centered grids

Compositional Simulation of the Vapex Process

Applications Of Dynamic Gridding To Thermal Simulations

Dynamic Grid Refinement and Amalgamation for Compositional Simulation

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