[01634] Mimetic schemes applied to the convection diffusion equation: A numerical comparison.

Session Time & Room : 2D (Aug.22, 15:30-17:10) @E702

Type : Contributed Talk

Abstract : Mimetic Finite Difference Schemes (DFM) are increasingly present in the numerical resolution of transient problems [1] since they are more precise than traditional Finite Difference (DF) schemes. However, there are methods in DF that use appropriate combinations of schemes in different nodes in order to eliminate the numerical spread of the method, [2]. In these cases, DF methods are more accurate than DFM. In this work, we start from the equation of convection-diffusion of an incompressible fluid
∂u/∂t+v·∇u=∇·(K∇u), (1)
where u(x, t) represents the unknown of the problem, v(x, t) is the velocity, K is the diffusion tensor, and DFM is defined that eliminates the numerical diffusion presented by traditional DFM schemes. To measure the effectiveness of the mimetic schemes, for different configurations of (1), they are compared with the equivalent schemes in DF with the same order of precision as the DFM; for this purpose, the second-order schemes proposed by [2, 3] are taken. Finally, different comparisons are made to verify the results obtained by the given schemes.
References:
[1] Castillo J. and Grone R.D. A matrix analysis approach to higher-order approximations for divergence and gradients satisfying a global conservation law. SIAM J. Matrix Anal. Appl., 25(1):128– 142, 2003.
[2] Mehdi Dehghan. Weighted finite difference techniques for the one-dimensional advection-diffusion equation. Applied Mathematics and Computation, 147(2):307–319, 2004.
[3] Mehdi Dehghan. Quasi-implicit and two-level explicit finite-difference procedures for solving the one-dimensional advection equation. Applied Mathematics and Computation, 167(1):46–67, 2005.