Multidimensional upwinding. II: Decomposition of the Euler equations into advection equations. (English) Zbl 0932.76051

Summary: Based on a genuine multidimensional numerical scheme, called the method of transport, we derive a form of the compressible Euler equations, capable of a linearization for any space dimension. This form enables a rigorous error analysis of the linearization error without the knowledge of the numerical method used to solve the linear equations. The generated error can be eliminated by special correction terms in the linear equations. Hence, existing scalar high-order methods can be used to solve the linear equations and obtain high-order accuracy in space and time for the nonlinear conservation law. In this approach, the scalar version of the method of transport is used to solve the linear equations. This method is multidimensional and reduces the solution of the partial differential equation to an integration process. Convergence histories presented at the end of the paper show that the numerical results agree with the theoretical predictions. \(\copyright\) Academic Press.


76M20 Finite difference methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)


Full Text: DOI


[1] Collela, P., Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys., 87, 171 (1990)
[2] Fey, M., Multidimensional upwinding. Part I. The method of transport for solving the Euler equations, J. Comput. Phys., 143, 159 (1998) · Zbl 0932.76050
[3] Fey, M.; Jeltsch, R.; Maurer, J.; Morel, A.-T., The Method of Transport for Nonlinear Systems of Hyperbolic Conservation Laws in Several Space Dimensions (1997)
[4] Fey, M.; Morel, A.-T., Multidimensional Method of Transport for the Sallow Water Equations (1995)
[5] Kröner, D., Numerical Schemes for Conservation Laws (1997) · Zbl 0872.76001
[6] Langseth, J. O.; LeVeque, R. J., Three-dimensional Euler computations using CLAWPACK, Conf. on Numer. Meth. for Euler and Navier-Stokes Eq. (1995)
[7] LeVeque, R. J., High resolution finite volume methods on arbitrary girds via wave propagation, J. Comput. Phys., 78, 36 (1988) · Zbl 0649.65050
[8] Maurer, J., The Method of Transport for Mixed Hyperbolic-Parabolic Systems (1997)
[9] Steger, J. L.; Warming, R. F., Flux vector splitting of the inviscid gasdynamic equations with application to finite difference methods, J. Comput. Phys., 40, 263 (1981) · Zbl 0468.76066
[10] Strang, G., On the construction and comparison of difference schemes, J. Numer. Anal., 5, 506 (1968) · Zbl 0184.38503
[11] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 115 (1984) · Zbl 0573.76057
[12] Yee, H., Upwind and Symmetric Shock-Capturing Schemes (1987)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.