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Wesseling , Feistauer et al. The FVM has also been very early introduced in the framework of oil reservoir engineering see Peaceman Such an expression of the flux is encountered when using e. Fourier's law heat conduction , Darcy's law flow in porous media or Fick's law chemical diffusion. This is the famous "two-point flux approximation" scheme, which has been widely used. It is simple, and also much appreciated because it respects a discrete version of the maximum principle; in particular, the approximate solution does not oscillate and stays within its physical bounds.

The convergence analysis of the FVM for diffusion equations in this particular case is thoroughly performed in Eymard et al. The error analysis relies on the conservativity and consistency of the numerical flux.

Volume 219, Number 2, October 2012

Unfortunately, all the linear schemes devised in the general case of unstructured meshes and general linear operators do not seem to satisfy a discrete version of the maximum principle; for severely distorted meshes or for highly anisotropic operators, the risk of oscillations and violation of the physical bounds is high. Ongoing research using in particular nonlinear schemes for linear problems is under way to avoid these drawbacks see the proceedings of Finite Volume for Complex Applications V. Nevertheless, the fluxes must be carefully approximated in order to ensure the stability and convergence of the scheme.

Provided that, for an explicit scheme, the so-called CFL condition be satisfied this condition prescribes a bound for the time step with respect to the size of the mesh , these schemes satisfy the following properties see Eymard et al. The Godunov scheme is a very efficient scheme when the above solution of the Riemann problem is easily constructed.

Applied Mathematics and Computation, Volume 219

When the solution of the Riemann problem becomes too complex or expensive, alternative schemes based on approximate Riemann solvers can be used see Eymard et al. See also these references for some extensions to multi-dimensional problems. Eymard, T. Finite volume methods. Ciarlet and J.

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North-Holland, Amsterdam, Eymard and J. Finite volumes for complex applications V. Wang, "Solution of the fractional Black-Scholes option pricing model by finite difference method," Abstract and Applied Analysis, vol. Atangana and A. Karatay and S.

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      Samko, A. Kilbas, and O. Jumarie, "Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions," Applied Mathematics Letters, vol. However, users may print, download, or email articles for individual use. CC BY. Some complexity on multiscale analysis can be simplified by the fractional order calculus. Preliminaries 2.

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      Integration with respect to dta. Let f x denote a continuous function; then the solution of 12 is defined by the equality f f t d? Existence and Uniqueness of the Nonnegative Solution Theorem 4. We remark that a solution which is not stable is said to be unstable. Theorem 7. Consider the following. Example 9.

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      Log out of ReadCube. This paper is concerned with accurate and efficient numerical methods for solving parabolic differential equations. The scheme is proved to be unconditionally stable. All computations are implemented in one direction and the CPU time is relatively smaller compared with some other compact computational schemes. Numerical results are presented to show the accuracy and efficiency of the new algorithm for the parabolic differential equations.

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