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The repair paradigm and application to conservation laws. (English) Zbl 1107.65341
Summary: Repair is a conservative, post-processing procedure to be used in numerical methods for hyperbolic conservation laws in order to preserve certain qualitative characteristics of the numerical solution, such as positivity of density and internal energy, by means of redistribution of conserved quantities such as mass, momentum and total energy among the cells of the mesh. In this paper we describe the repair paradigm and prove several theorems which form a theoretical foundation for the repair procedures. We consider two applications of repair and present corresponding numerical results. The first application deals with improving properties of the remapping (conservative interpolation) stage of arbitrary Lagrangian–Eulerian (ALE) methods for the gas dynamics equations, in which the solution is conservatively transferred from one mesh to another. One requirement for remapping is that the interpolated density and internal energy on the new mesh have to stay positive. Another desirable property is that the remapping procedure should not create new extrema for the velocity field. For various reasons these properties may not be satisfied, especially for high-order methods. Repair plays a supplemental role by bringing gas dynamics quantities into physically justified bounds. Another application of repair is to improve the quality of numerical methods for advection of some scalar tracer field with prescribed divergence-free velocity field, in which case the advection equation can be written as a conservation law, and therefore the total amount of tracer is conserved. We show how the repair procedure allows us to reduce oscillations in a numerical solution obtained by a formally high-order method. Repair offers an alternative to more classical methods of reducing oscillations and maintaining positivity.

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
80M25 Other numerical methods (thermodynamics) (MSC2010)
80A20 Heat and mass transfer, heat flow (MSC2010)
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