WSEAS Transactions on Applied and Theoretical Mechanics
Print ISSN: 1991-8747, E-ISSN: 2224-3429
Volume 7, 2012
Explicit and Implicit TVD High Resolution Schemes in 2D
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Abstract: The present work compares the TVD schemes of Roe, of Van Leer, of Yee,Warming and Harten, of Harten, of Yee and Kutler and of Hughson and Beran applied to the solution of an aeronautical problem. Only the Van Leer scheme is a flux vector splitting one. The others are of flux difference splitting type. The Roe and Van Leer schemes reach second order accuracy and TVD properties by the use of a MUSCL approach, which employs five different types of nonlinear limiters, that assures TVD properties, being them: Van Leer limiter, Van Albada limiter, minmod limiter, Super Bee limiter and -limiter. The other schemes are based on the Harten’s ideas of the construction of a modified flux function to obtain second order accuracy and TVD characteristics. The implicit schemes employ an ADI (“Alternating Direction Implicit”) approximate factorization to solve implicitly the Euler equations, whereas in the explicit case a time splitting method is used. Explicit and implicit results are compared trying to emphasize the advantages and disadvantages of each formulation. The Euler equations in conservative form, employing a finite volume formulation and a structured spatial discretization, are solved in two-dimensions. The steady state physical problem of the supersonic flow along a compression corner is studied. A spatially variable time step procedure is employed aiming to accelerate the convergence of the numerical schemes to the steady state condition. This technique has proved an excellent behavior in terms of convergence gains, as shown in Maciel. The results have demonstrated that the most accurate solutions are provided by the Roe TVD scheme in its Super Bee variant.
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Keywords: Roe scheme, Van Leer scheme, Yee, Warming and Harten scheme, Harten scheme, Yee and Kutler scheme, Hughson and Beran scheme, Explicit and implicit formulations, TVD formulation, Euler and Navier-Stokes equations