TY - JOUR

T1 - The finite volume-complete flux scheme for advection-diffusion-reaction equations

AU - Thije Boonkkamp, ten, J.H.M.

AU - Anthonissen, M.J.H.

PY - 2011

Y1 - 2011

N2 - We present a new finite volume scheme for the advection-diffusion-reaction equation. The scheme is second order accurate in the grid size, both for dominant diffusion and dominant advection, and has only a three-point coupling in each spatial direction. Our scheme is based on a new integral representation for the flux of the one-dimensional advection-diffusion-reaction equation, which is derived from the solution of a local boundary value problem for the entire equation, including the source term. The flux therefore consists of two parts, corresponding to the homogeneous and particular solution of the boundary value problem. Applying suitable quadrature rules to the integral representation gives the complete flux scheme. Extensions of the complete flux scheme to two-dimensional and time-dependent problems are derived, containing the cross flux term or the time derivative in the inhomogeneous flux, respectively. The resulting finite volume-complete flux scheme is validated for several test problems.

AB - We present a new finite volume scheme for the advection-diffusion-reaction equation. The scheme is second order accurate in the grid size, both for dominant diffusion and dominant advection, and has only a three-point coupling in each spatial direction. Our scheme is based on a new integral representation for the flux of the one-dimensional advection-diffusion-reaction equation, which is derived from the solution of a local boundary value problem for the entire equation, including the source term. The flux therefore consists of two parts, corresponding to the homogeneous and particular solution of the boundary value problem. Applying suitable quadrature rules to the integral representation gives the complete flux scheme. Extensions of the complete flux scheme to two-dimensional and time-dependent problems are derived, containing the cross flux term or the time derivative in the inhomogeneous flux, respectively. The resulting finite volume-complete flux scheme is validated for several test problems.

U2 - 10.1007/s10915-010-9388-8

DO - 10.1007/s10915-010-9388-8

M3 - Article

VL - 46

SP - 47

EP - 70

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

IS - 1

ER -