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Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems

Summary: Construction of second order accurate monotone and
stable residual distribution schemes for unsteady flow problems
Reemi Abgrall *,1
, Mohamed Mezine 2
Matheematiques Appliqueees de Bordeaux, Universitee Bordeaux I, 351 cours de la Libeeration, 33 405 Talence Cedex, France
Received 22 May 2002; received in revised form 15 October 2002; accepted 26 November 2002
The aim of this paper is to construct upwind residual distribution schemes for the time accurate solution of hy-
perbolic conservation laws. To do so, we evaluate a space­time fluctuation based on a space­time approximation of the
solution and develop new residual distribution schemes which are extensions of classical steady upwind residual dis-
tribution schemes. This method has been applied to the solution of scalar advection equation and to the solution of the
compressible Euler equations both in two space dimensions. The first version of the scheme is shown to be, at least in its
first order version, unconditionally energy stable and possibly conditionally monotonicity preserving. Using an idea of
Csik et al. [Space­time residual distribution schemes for hyperbolic conservation laws, 15th AIAA Computational
Fluid Dynamics Conference, Anahein, CA, USA, AIAA 2001-2617, June 2001], we modify the formulation to end up
with a scheme that is unconditionally energy stable and unconditionally monotonicity preserving. Several numerical
examples are shown to demonstrate the stability and accuracy of the method.
Ó 2003 Elsevier Science B.V. All rights reserved.
AMS: 65C99; 65M60; 76N10
Keywords: Compressible flow solvers; Residual schemes; Fluctuation splitting schemes; Euler equations; Unsteady flows; Unstructured


Source: Abgrall, Rémi - Institut de Mathematiques de Bordeaux, Université Bordeaux


Collections: Mathematics