Total of time variation diminishing principle for conservation laws

Abstract

We introduce a new principle for devising numerical schemes for conservation laws in one and multiple dimensions. The new formulation is based on the Total of Time Variation (TOTV) defined as the volume integral of the magnitude of the time derivative. For the one-dimensional scalar advection equation with a constant velocity, TOTV and the usual total variation (TV) are the same except for a constant factor. For non-linear equations and/or in multiple dimensions, TV and TOTV are different. We show that TOTV is a conserved quantity for one- and multi-dimensional scalar conservation laws with a non-linear flux function that can depend on the spatial coordinates as well. We call a numerical scheme that ensures that the discrete form of TOTV is not increasing in time a Total of Time Variation Diminishing (TOTVD) method. A TOTVD scheme is stable against catastrophic instabilities that would lead to uncontrolled growth of the time derivative. We show that the first order upwind scheme with a finite time step satisfying the usual CFL condition is TOTVD for all equations that satisfy the TOTVD property analytically. We demonstrate the difference between TV and TOTV with numerical tests.