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Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics. (English) Zbl 1280.76030
Summary: In a pair of earlier papers the author showed the importance of divergence-free reconstruction in adaptive mesh refinement problems for magnetohydrodynamics (MHD) and the importance of the same for designing robust second order schemes for MHD. Second order accurate divergence-free schemes for MHD have shown themselves to be very useful in several areas of science and engineering. However, certain computational MHD problems would be much benefited if the schemes had third and higher orders of accuracy. In this paper we show that the reconstruction of divergence-free vector fields can be carried out with better than second order accuracy. As a result, we design divergence-free weighted essentially non-oscillatory (WENO) schemes for MHD that have order of accuracy better than second. A multi-stage Runge-Kutta time integration is used to ensure that the temporal accuracy matches the spatial accuracy. While this is achieved quite simply up to third order in time, going beyond third order is most simply achieved by using the ADER-WENO schemes that are detailed in a companion paper. (ADER stands for Arbitrary Derivative Riemann Problem.) Accuracy analysis is carried out and it is shown that the schemes meet their design accuracy for smooth problems. Stringent tests are also presented showing that the schemes perform well on those tests.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76W05 Magnetohydrodynamics and electrohydrodynamics
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