Compact High Order Complete Flux Schemes
In this paper we outline the complete flux scheme for an advection-diffusion-reaction model problem. The scheme is based on the integral representation of the flux, which we derive from a local boundary value problem for the entire equation, including the source term. Consequently, the flux consists of a homogeneous part, corresponding to the advection-diffusion operator, and an inhomogeneous part, taking into account the effect of the source term. We apply (weighted) Gauss quadrature rules to derive the standard complete flux scheme, as well as a compact high order variant. We demonstrate the performance of both schemes.
- 1.M.J.H. Anthonissen, J.H.M. ten Thije Boonkkamp, A compact high order finite volume scheme for advection-diffusion-reaction equations, in Numerical Analysis and Applied Mathematics: International Conference on Numerical Analysis and Applied Mathematics 2009, AIP Conference Proceedings, vol. 1168 (2009), pp. 410–414Google Scholar
- 2.N. Kumar, J.H.M. ten Thije Boonkkamp, B. Koren, A. Linke, A nonlinear flux approximation scheme for the viscous Burgers’ equation, in Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems, ed. by C. Cancès et al. Springer Proceedings in Mathematics & Statistics, vol. 200 (2017), pp. 457–465Google Scholar
- 3.K.W. Morton, Numerical Solution of Convection-Diffusion Problems. Applied Mathematics and Mathematical Computation, vol. 12 (Chapman & Hall, London, 1996)Google Scholar
- 5.J.H.M. ten Thije Boonkkamp, M.J.H. Anthonissen, R.J. Kwant, A two-dimensional complete flux scheme in local flow adapted coordinates, in Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems, ed. by C. Cancès et al. Springer Proceedings in Mathematics & Statistics, vol. 200 (2017), pp. 437–445Google Scholar