An adaptive, hanging-node, discontinuous isogeometric analysis method for the first-order form of the neutron transport equation with discrete ordinate (SN) angular discretisation

In this paper a discontinuous, hanging-node, isogeometric analysis (IGA) method is developed and applied to the first-order form of the neutron transport equation with a discrete ordinate (SN) angular discretisation in two-dimensional space. The complexities involved in upwinding across curved element boundaries that contain hanging-nodes have been addressed to ensure that the scheme remains conservative. A robust algorithm for cycle-breaking has also been introduced in order to develop a unique sweep ordering of the elements for each discrete ordinates direction. The convergence rate of the scheme has been verified using the method of manufactured solutions (MMS) with a smooth solution. Heuristic error indicators have been used to drive an adaptive mesh refinement (AMR) algorithm to take advantage of the hanging-node discretisation. The effectiveness of this method is demonstrated for three test cases. The first is a homogeneous square in a vacuum with varying mean free path and a prescribed extraneous unit source. The second test case is a radiation shielding problem and the third is a 3×3 “supercell” featuring a burnable absorber. In the final test case, comparisons are made to the discontinuous Galerkin finite element method (DGFEM) using both straight-sided and curved quadratic finite elements

Boundary layers in weak solutions to hyperbolic conservation laws

This paper is concerned with the initial-boundary value problem for a nonlinear hyperbolic system of conservation laws. We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing viscosity method and finite difference schemes (Lax-Friedrichs type schemes, Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming solely uniform L\infty bounds on the approximate solutions and so dealing with L\infty solutions, we derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary Young measure reduces to a Dirac mass. Form the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not. Each formulation is based a set of admissible boundary values, following Dubois and LeFloch's terminology in ``Boundary conditions for nonlinear hyperbolic systems of conservation laws'', J. Diff. Equa. 71 (1988), 93--122. The local structure of those sets and the well-posedness of the corresponding initial-boundary value problem are investigated. The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics.Comment: 43 page

Time--Splitting Schemes and Measure Source Terms for a Quasilinear Relaxing System

Several singular limits are investigated in the context of a $2 \times 2$ system arising for instance in the modeling of chromatographic processes. In particular, we focus on the case where the relaxation term and a $L^2$ projection operator are concentrated on a discrete lattice by means of Dirac measures. This formulation allows to study more easily some time-splitting numerical schemes

Coupling techniques for nonlinear hyperbolic equations. IV. Multi-component coupling and multidimensional well-balanced schemes

This series of papers is devoted to the formulation and the approximation of coupling problems for nonlinear hyperbolic equations. The coupling across an interface in the physical space is formulated in term of an augmented system of partial differential equations. In an earlier work, this strategy allowed us to develop a regularization method based on a thick interface model in one space variable. In the present paper, we significantly extend this framework and, in addition, encompass equations in several space variables. This new formulation includes the coupling of several distinct conservation laws and allows for a possible covering in space. Our main contributions are, on one hand, the design and analysis of a well-balanced finite volume method on general triangulations and, on the other hand, a proof of convergence of this method toward entropy solutions, extending Coquel, Cockburn, and LeFloch's theory (restricted to a single conservation law without coupling). The core of our analysis is, first, the derivation of entropy inequalities as well as a discrete entropy dissipation estimate and, second, a proof of convergence toward the entropy solution of the coupling problem.Comment: 37 page

Improvements to embedded shock wave calculations for transonic flow-applications to wave drag and pressure rise predictions

The numerical solution of a single, mixed, nonlinear equation with prescribed boundary data is discussed. A second order numerical procedure for solving the nonlinear equation and a shock fitting scheme was developed to treat the discontinuities that appear in the solution