A variational splitting of high-order linear multistep methods for heat transfer and advection–diffusion parabolic problems

Abstract

The main result of this paper is the generalization of the concept of variational splitting proposed by Behnoudfar et al. (2018) into higher-order time-stepping schemes. The key novelty of the technique proposed here is the following: when using the isogeometric finite elements, one can factorize the matrix of the linear system and obtain the linear complexity. This matrix has the form M+ηK, where M is the mass matrix, K is the stiffness matrix. Following the technique of Behnoudfar et al. (2018), and factorizing M+ηK one would get the additional error of order η2. We propose the technique to couple the possibility of factorization of M+ηK with the higher-order time-stepping schemes and preserve this higher order. Thus, we introduce a variational splitting for implicit, linear multistep methods of order s for parabolic partial differential equations. Our technique exploits the tensor-product structure of the discretization in space (mesh and basis functions) to derive an update strategy with a linear cost regarding the total number of degrees of freedom for multi-dimensional problems. As an example, we consider two popular methods: Adams–Moulton (AM) and backward differentiation formulae (BDF). We derive the corresponding coefficients for AM and BDF methods. We present a C++ implementation of the time-integration schemes, targeting shared-memory Linux clusters. We present numerical results for the heat transfer problem and advection–diffusion problem. We also discuss the limitations of the method and possible extensions overcoming this limitation by using our method as a preconditioner for an iterative solver.