Abstract
Second order explicit discretization methods for convection dominated transport are studied here from the point of view of discrete minimum and maximum property of numerical solutions. These methods are based on vertex-centered finite volume methods on general unstructured computational grids. It will be shown that “standard TVD methods” [13],[14] do not fulfill in general the discrete minimum and maximum property and that these methods must be modified to obtain numerical solutions with no unphysical oscillations. Finally, new methods based on our theoretical results are proposed.
This work is funded by the Federal Ministry of Economics and Technology (BMWi) under the contract number 02 E 9148 2
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References
Bastian, P., Birken, K., Johannsen, K., Lang, S., Neuss, N., Rentz-Reichert, H.: UG-a flexible software toolbox for solving partial differential equations. Computing and Visualization in Science 1(1) (1997) 27–40.
Ciarlet, P.G.: The Finite Element Methods for Elliptic Problems. North Holland Publishing Company, Amsterdam (1978)
Courant, R., Friedrics, K.O., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Annalen 100 (1928) 32–74.
Fein, E., Kühle, T., Noseck, U.: Entwicklung eines Programms zur dreidimensionalen Modellierung des Schadstofftransportes. Fachliches Feinkonzept, Braunschweig (2001)
Frolkovič, P.: Maximum principle and local mass balance for numerical solutions of transport equation coupled with variable density flow. Acta Mathematica Universitatis Comenianae 1 (67) (1998) 137–157.
Frolkovič, P., Geiser, J.: Numerical Simulation of Radionuclides Transport in Double Porosity Media with Sorption. Conference of Scientific Computing, Proceedings of Algorithmy (2000) 28–36.
Frolkovič, P.: Flux-based method of characteristics for contaminant transport in flowing groundwater. Computing and Visualization in Science (2002) (to appear)
Geiser, J.: Numerical Simulation of a Model for Transport and Reaction of Radionuclides. In: Margenov, S., Wasniewski, J., and Yalamov, P. (eds.): Large-Scale Scientific Computing. Lecture Notes in Computer Science, Vol. 2179, Springer-Verlag (2001) 487–496.
Ikeda, T.: Maximum principle in finite element models for convection-diffusion phenomena. North-Holland-Publishing Company, Amsterdam.
Johannsen, K.: An Aligned 3D-Finite-Volume Method for Convection-Diffusion Problems. In: Helmig, R., Jäger, W., Kinzelbach, W., Knabner, P., Wittum, G. (eds.): Modeling and Computation in Environmental Sciences. Vieweg, Braunschweig, 59 (1997) 227–243.
Klöfkorn, R.: Simulation von Abbau-und Transportprozessen gelöster Schadstoffe im Grundwasser. Diplomarbeit, Institut für Angewandte Mathematik, Universität Freiburg (2001)
Klöfkorn, R., Kröner, D., Ohlberger, M.: Local adaptive methods for convection dominated problems. Internat. J. Numer. Methods Fluids (2002) (to appear)
Sonar, Th.: On the design of an upwind scheme for compressible flow on general triangulation. Numerical Analysis, 4 (1993) 135–148.
Sonar, Th.: Mehrdimensionale ENO-Verfahren. Advances in numerical mathematics, Teubner Stuttgart (1997)
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Frolkovič, P., Geiser, J. (2003). Discretization Methods with Discrete Minimum and Maximum Property for Convection Dominated Transport in Porous Media. In: Dimov, I., Lirkov, I., Margenov, S., Zlatev, Z. (eds) Numerical Methods and Applications. NMA 2002. Lecture Notes in Computer Science, vol 2542. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36487-0_50
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