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Directional split is a common technique used in Finite–Volume Methods (FVM) for solving multidimensional hyperbolic partial differential equations on regular grids. The directional split method is based on independent one–dimensional flux calculation for each Cartesian direction. Second order accuracy is achieved by doing a piecewise linear calculation of cell quantities in a Monotonic Upstream–Centered Scheme for Conservation Laws (MUSCL) scheme. This one–dimensional reconstruction results in artificial grid–orientation dependence of the numerical solution, i.e. some anisotropy are introduced even for originally isotropic problems. To mitigate the anisotropy of these schemes we propose to use extended two–dimensional stencils during the reconstruction step, while keeping the directional–split nature of the scheme. In this work, we introduce a novel multi–dimensional modification of the minmod slope limiter in the context of a MUSCL Hancock directional split FVM numerical scheme. Local direction of the unlimited gradient is utilized in the construction of our limiter. Improvements in the isotropy of the numerical solution are demonstrated for the linear advection as well as for standard hydrodynamics test cases on square meshes.

ISOTROPY–PRESERVING SLOPE LIMITERS FOR FINITE VOLUME METHODS ON SQUARE MESHES