# Semi-implicit methods based on inflow implicit and outflow explicit time discretization of advection

## Main Article Content

## Abstract

We introduce several numerical methods for the solution of advection equation using semi-implicit time discretization in which the inflow fluxes are discretized implicitly and the outflow fluxes explicitly. We derive the so called $\kappa$-scheme and show it is $2^{nd}$ order accurate and uncon\-ditionally stable in 1D and 2D case for tensor grids with a special choice of $\kappa$ giving $3^{rd}$ order accurate scheme for constant speed in 1D. Moreover, we present a $2^{nd}$ order accurate and unconditionally stable Corner Transport scheme in 2D case for tensor grids that is $3^{rd}$ order accurate for constant velocity. We discuss several improved properties of these schemes when compared to analogous fully explicit and fully implicit schemes.

## Article Details

How to Cite

FROLKOVIČ, Peter.
Semi-implicit methods based on inflow implicit and outflow explicit time discretization of advection.

**Proceedings of the Conference Algoritmy**, [S.l.], p. 165-174, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/405>. Date accessed: 22 sep. 2017.
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Articles

## References

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[17] H. Zhao. A fast sweeping method for eikonal equations. Math. Comput., 74(250):603–628, 2004.

[2] P. Frolkovič, M. Lampe, and G. Wittum. Numerical simulation of contaminant transport in groundwater using software tools r3 t. Comp. Vis. Sci., to appear.

[3] P. Frolkovič and K. Mikula. High-resolution flux-based level set method. SIAM J. Sci. Comp., 29(2):579–597, 2007.

[4] P. Frolkovič, K. Mikula, and J. Urb ́ an. Semi-implicit finite volume level set method for advective motion of interfaces in normal direction. Appl. Num. Math., 95:214–228, September 2015.

[5] P. Frolkovič and C. Wehner. Flux-based level set method on rectangular grids and computations of first arrival time functions. Comp. Vis. Sci., 12(5):297–306, 2009.

[6] B.P. Leonard. A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comput. Methods in Appl. Mech. Eng., 19(1):59–98, 1979.

[7] R. J. LeVeque. High resolution finite volume methods on arbitrary grids via wave propagation. J. Comput. Phys., 78(1):36–63, 1988.

[8] R. J. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge UP, 2002.

[9] K. Mikula and M. Ohlberger. Inflow-Implicit/Outflow-Explicit scheme for solving advection equations. In J. Fort et al., editor, Finite Volumes for Complex Applications VI, pages 683–692. Springer Verlag, 2011.

[10] K. Mikula, M. Ohlberger, and J. Urbán. Inflow-implicit/outflow-explicit finite volume methods for solving advection equations. Appl. Numer. Math., 85:16–37, 2014.

[11] B. Seibold. Numerical methods for partial differential equations. Lecture notes at ocw.mit.edu/courses/mathematics/18-336-numerical-methods-for-partial-differentialequations-spring-2009, Massachusetts Institute of Technology, 2009.

[12] E. F. Toro. Riemann solvers and numerical methods for fluid dynamics: A practical introduction. Springer, 2009.

[13] L.N. Trefethen. Finite difference and spectral methods for ordinary and partial differential equations. Cornell University, 1996.

[14] B. Van Leer. Towards the ultimate conservative difference scheme III. Upstream-centered finitedifference schemes for ideal compressible flow. J. Comput. Phys., 23(3):263–275, 1977.

[15] P. Wesseling. Principles of Computational Fluid Dynamics. Springer Series in Computational Mathematics, 2001.

[16] Wolfram Research, Inc. Mathematica, Version 10.2. Champaign, IL, 2015.

[17] H. Zhao. A fast sweeping method for eikonal equations. Math. Comput., 74(250):603–628, 2004.