A very high-order finite volume method for the time-dependent convection–diffusion problem with Butcher Tableau extension

• Published in: Computers & mathematics with applications (1987) Vol. 68; no. 10; pp. 1292 - 1311
• Main Authors: Clain, S., Machado, G.J.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.11.2014
• Subjects: Accuracy; Butcher Tableau; Computer simulation; Convection–diffusion; Discretization; Finite volume; Finite volume method; Mathematical models; Polynomial reconstruction; Polynomials; Runge-Kutta method; Very high-order; Finite volume; Convection–diffusion; Very high-order; Polynomial reconstruction; Butcher Tableau
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2014.09.001

The time discretization of a very high-order finite volume method may give rise to new numerical difficulties resulting into accuracy degradations. Indeed, for the simple one-dimensional unstationary convection–diffusion equation for instance, a conflicting situation between the source term time discretization and the boundary conditions may arise when using the standard Runge–Kutta method. We propose an alternative procedure by extending the Butcher Tableau to overcome this specific difficulty and achieve fourth-, sixth- or eighth-order of accuracy schemes in space and time. To this end, a new finite volume method is designed based on specific polynomial reconstructions for the space discretization, while we use the Extended Butcher Tableau to perform the time discretization. A large set of numerical tests has been carried out to validate the proposed method.