I’m writing a paper right now that reviews some suggestions for solving the ubiquitous scalar advection equation,

$\partial_t u + \partial_x u = 0$,

which is the basis for remap in arbitrary Lagrangian-Eulerian codes and more generally hyperbolic PDEs.  I will post different sections from the paper week to week as I work toward the submission of the manuscript in a month.  A lot of the paper is already written, so the posts will show the progression of polished text.

The standard method for solving scalar advection has been Van Leer’s piecewise linear method (with an appropriate limiter), and its generalizations.   The method is based on polynomial interpolation normalized within a cell

$\xi =\frac{(x-x_j)}{\Delta x};P(\theta)=P_0 + P_1 \xi$, where $P_0=u_j; P_1 = \frac{\partial u}{\partial x}\Delta x$.

Van Leer’s method is really only one of the many schemes he suggested in his 1977 paper with the other methods being largely ignored (with the exception of the PPM method).

My paper will review Van Leer’s schemes and the analysis from the 1977 paper and then suggest how these concepts can be extended.  Some of the methods have direct connections to discontinuous Galerkin, but others are unique.  Some of these methods are completely different than anything else out there and worth a lot of attention.  These methods have the character of both finite volume and finite difference method plus have compactness that might be quite advantageous on modern computers.  Their numerical error characteristics are superior to the standard methods.

Stay tuned.