Commun. Comput. Phys., 9 (2011), pp. 688-710.

Constraint Preserving Schemes Using Potential-Based Fluxes I. Multidimensional Transport Equations

Siddhartha Mishra 1, Eitan Tadmor 2*

1 Centre of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, N-0316 Oslo, Norway.
2 Center of Scientific Computation and Mathematical Modeling (CSCAMM), and Department of Mathematics, and Institute for Physical Sciences and Technology (IPST), University of Maryland, MD 20742-4015, USA.

Received 3 September 2009; Accepted (in revised version) 9 November 2009
Available online 17 September 2010


We consider constraint preserving multidimensional evolution equations. A prototypical example is provided by the magnetic induction equation of plasma physics. The constraint of interest is the divergence of the magnetic field. We design finite volume schemes which approximate these equations in a stable manner and preserve a discrete version of the constraint. The schemes are based on reformulating standard edge centered finite volume fluxes in terms of vertex centered potentials. The potential-based approach provides a general framework for faithful discretizations of constraint transport and we apply it to both divergence preserving as well as curl preserving equations. We present benchmark numerical tests which confirm that our potential-based schemes achieve high resolution, while being constraint preserving.

AMS subject classifications: 65M06, 35L65

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Key words: Multidimensional evolution equations, magnetohydrodynamics, finite-difference, finite volume schemes, constraint transport, potential-based schemes.

*Corresponding author.
Email: (S. Mishra), (E. Tadmor)

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