Space-Time Integrated Least-Squares: Solving a Pure Advection Equation with a Pure Diffusion Operator

An alternative formulation for multidimensional scalar advection is derived following both a conservative and a variational approach, by applying the least-squares method simply generalized to the space-time domain. In the space-time framework pure advection is regarded as a process involving only anisotropic diffusion along space-time characteristics. The resulting parabolic-type equation lends itself to a straightforward Galerkin integration that yields a symmetric, diagonally dominant, positive, and unconditionally stable operator. The conditions of equivalence between the advective problem and its parabolized counterpart are established by using standard variational theory in anisotropic Sobolev spaces specially designed for advection equations. To demonstrate the general applicability of the method, "parabolized advection" is simulated in 2D manifolds embedded in 3D and 4D space-time domains.