A modified linear discontinuous spatial discretization method in planar geometry

Abstract

The standard derivation of the linear discontinuous (LD) spatial discretization method for planar geometry transport problems is modified to produce an entire family of methods. This family is characterized by a real parameter B which assumes the value B = 3 in the usual LD method. This discretization scheme is investigated for positivity and accuracy as a function of 0 . Numerical results are compared with exact analytic solutions for the simple rod model of transport theory.

A comparison of numerical schemes for transport problems

“…A necessary condition for avoiding oscillatory behaviour and negative fluxes in the discrete solution (17) [7], which probably explains the 'robustness' observed by Larsen and Morel [5] when using the MLD scheme.…”

“…To wit, let us define the inner product, and establish a grid of discrete points {xj) for the coordinate x (for concreteness, we take xo = 0). The values of the coefficients +j,j # 0, are determined by (8) given the initial value As shown by Szilard and Pomraning [7], Eq. (10) defines a valid finite difference scheme for any value of 6, and reduces to the standard DD scheme when 6 = 0.…”

“…(7), but this time we test with the same teepee functions. This leads to the system of equations given by Notice that the scheme requires three points in the site index j and so, unlike the DD, LD, and MLD schemes, must be initialized with $0 and an additional condition on the coefficients { $ j } .…”