VIP (Vector Image Polygon) multi-dimensional slope limiters for scalar variables

A recently formulated frame-invariant monotonicity criterion and slope limiter for vectors was applied to the Staggered Mesh Godunov-SMG/Q scheme for Lagrangian and ALE hydrodynamics. The VIP (vector-space polygon or polyhedron) was shown to be a natural extension of monotonicity constraints from scalar to vector variables. Taking notice of the fact that gradients of scalars are vectors, we now seek to use this new concept to devise better, and perhaps truly multidimensional, slope limiters for scalar variables. The proposed scheme constitutes a generalization of a 1D monotonic-averaging limiter for scalars, to a VIP type limiter for (vector) gradients of scalars in 2D or 3D. Test cases computed by the SMG/ALE scheme, using the new VIP limiter for gradients, are presented. As we can see from them, the new method, while being robust in strong shock computations, can better preserve gradients of density under advection.