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Finite difference
About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.
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TL;DR: In this paper, two numerical models for simulating three-dimensional, two-phase fluid flow and heat transport in geothermal reservoirs are introduced, one based on a 3D formulation of the governing equations for geothermal reservoir and the other based on partial integration of the 3D equations in the vertical dimension.
Abstract: Two numerical models are introduced for simulating three-dimensional, two-phase fluid flow and heat transport in geothermal reservoirs. The first model is based on a three-dimensional formulation of the governing equations for geothermal reservoirs. Since the resulting two partial differential equations, posed in terms of fluid pressure and enthalpy, are highly nonlinear and inhomogeneous, they require numerical solution. The three-dimensional numerical model uses finite difference approximations, with fully implicit Newton-Raphson treatment of nonlinear terms and a block (vertical slice) successive iterative technique for matrix solution. Newton-Raphson treatment of nonlinear terms permits the use of large time steps, while the robust iterative matrix method reduces computer execution time and storage for large three-dimensional problems. An alternative model is derived by partial integration (in the vertical dimension) of the three-dimensional equations. This second model explicitly assumes vertical equilibrium (gravity segregation) between steam and water and can be applied to reservoirs with good vertical communication. The resulting equations are posed in terms of depth-averaged pressure and enthalpy and are solved by a two-dimensional finite difference model that uses a stable sequential solution technique, direct matrix methods, and Newton-Raphson iteration on accumulation and source terms. The quasi-three-dimensional areal model should be used whenever possible, because it significantly reduces computer execution time and storage and it requires less data preparation. The areal model includes effects of an inclined, variable-thickness reservoir and mass and energy leakage to confining beds. The model works best for thin (<500 m) reservoirs with high permeability. It can also be applied to problems with vertical to horizontal anisotropy when permeability is sufficiently high. Comparisons between finite difference and higher-order finite, element approximations show some advantage in using finite element techniques for single-phase problems. In general, for nonlinear two-phase problems the finite element method requires use of upstream weighting and diagonal lumping of accumulation terms. These lead to lower-order approximations and tend to obviate any advantage of using the finite element method.
87 citations
TL;DR: A general digital signal processing (DSP)-oriented framework where the functional equivalence of these two approaches is systematically elaborated and the conditions of building mixed models are studied.
Abstract: Digital waveguides and finite difference time domain schemes have been used in physical modeling of spatially distributed systems. Both of them are known to provide exact modeling of ideal one-dimensional (1D) band-limited wave propagation, and both of them can be composed to approximate two-dimensional (2D) and three-dimensional (3D) mesh structures. Their equal capabilities in physical modeling have been shown for special cases and have been assumed to cover generalized cases as well. The ability to form mixed models by joining substructures of both classes through converter elements has been proposed recently. In this paper, we formulate a general digital signal processing (DSP)-oriented framework where the functional equivalence of these two approaches is systematically elaborated and the conditions of building mixed models are studied. An example of mixed modeling of a 2D waveguide is presented.
87 citations
Proceedings Article•
06 Mar 2014TL;DR: A mixture of numerical methods including finite difference and body fitted co-ordinates are used to form a robust stable numerical scheme to solve the investment lag model presented in the paper by Bar-Ilan and Strange (1996).
Abstract: In this paper we use a mixture of numerical methods including finite difference and body fitted co-ordinates
to form a robust stable numerical scheme to solve the investment lag model presented in the paper by Bar-Ilan
and Strange (1996). This allows us to apply our methodology to models with different stochastic processes
that does not have analytic solutions.
87 citations
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TL;DR: In this paper, a new approach to the block SOR method applied to linear systems of equations which can be written as a matrix equation AX−XB=C was proposed.
87 citations
TL;DR: In this article, the total variation diminishing (TVD) finite difference scheme was interpreted as a Lax-Wendroff scheme plus an upwind weighted artificial dissipation term, which can be added to existing MacCormack method codes.
Abstract: In this paper we show that the total variation diminishing (TVD) finite difference scheme which was analysed by Sweby [8] can be interpreted as a Lax—Wendroff scheme plus an upwind weighted artificial dissipation term. We then show that if we choose a particular flux limiter and remove the requirement for upwind weighting, we obtain an artificial dissipation term which is based on the theory of TVD schemes, which does not contain any problem dependent parameters and which can be added to existing MacCormack method codes. Finally, we conduct numerical experiments to examine the performance of this new method.
86 citations