Improving Monotonicity of the 2 nd Order Backward Difference Time Integration Scheme by Temporal Limiting

Abstract

This work focuses on the behaviour of the 2 nd order accurate backward difference time integration scheme (BDF2) when used to solve unsteady hyperbolic conservation laws, including the compressible Euler equations, in the presence of strong gradients. This scheme is 2 nd order, A-stable but not monotonicity preserving [Ferziger and Peric(2002)]. The objective of this work is therefore to develop a time integration scheme based on BDF2 that preserves the monotonicity of the solution for moderately high CFL numbers \( (CFL = \mathcal{O}(10)) \) when used in combination with spatial TVD discretizations. We address the problem of monotone time integration using limiting techniques inspired from the finite-volume space discretizations methods. Similarly, we limit (in time) by blending the BDF2 time-integration scheme with a 1 st order accurate, A-stable and positive (and therefore monotonicity preserving) scheme. The temporal scheme described is independent from the spatial discretization scheme and from the TVD limiter used, as well as from the spatial computational stencil.