Efficient higher order single step methods for parabolic problems. I

Higher-Order Difference and Higher-Order Splitting Methods for 2D Parabolic Problems with Mixed Derivatives

In this article we discuss a combination between fourth-order finite difference methods and fourth-order splitting methods for 2D parabolic problems with mixed derivatives. The finite difference methods are based on higher-order spatial discretization methods, whereas the timediscretization methods are higher-order discretizations using CrankNicolson or BDF methods. The splitting methods are hi...

Single Step Galerkin Approximations for Parabolic Problems

In this paper we construct and analyze classes of single step methods of arbitrary order for homogeneous linear initial boundary value problems for parabolic equations with time-independent coefficients. The spatial discretization is done by means of general Galerkin-type methods.

Boundary Value Problems for Higher Order Parabolic Equations

We consider a constant coefficient parabolic equation of order 2m and establish the existence of solutions to the initial-Dirichlet problem in cylindrical domains. The lateral data is taken from spaces of Whitney arrays which essentially require that the normal derivatives up to order m−1 lie in L2 with respect to surface measure. In addition, a regularity result for the solution is obtained if...

Higher-order operator splitting methods for deterministic parabolic equations

Unified analysis of higher-order finite volume methods for parabolic problems on quadrilateral meshes

Finite volume methods (FVMs) have been widely used in scientific computing and engineering due to their easy implementation and the local conservation property. Lower-order FVMs are tightly related to finite difference or finite element methods, and have been extensively studied for a long time; see, e.g., Angelini et al. (2013), Bank & Rose (1987), Chatzipantelidis et al. (2008), Chou & Ye (20...