Book review: Principles of Computational Fluid Dynamics

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Book review: Mathematical Modelling in Continuum Mechanics

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Book review: Introduction to partial differential equations

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A Generalised Complete Flux scheme for anisotropic advection-diffusion equations

In this paper, we consider separating the discretisation of the diffusive and advective fluxes in the complete flux scheme. This allows the combination of several discretisation methods for the homogeneous flux with the complete flux (CF) method. In particular, we explore the combination of the hybrid mimetic mixed (HMM) method and the CF method, in order to utilize the advantages of each of these methods. The usage of HMM allows us to handle anisotropic diffusion tensors on generic polygonal (polytopal) grids; whereas the CF method provides a framework for the construction of a uniformly second order method, even when the problem is advection dominated

Illumination freeform design using Monge-Ampère equations

\u3cp\u3eAs a generic model for freeform optical systems, we combine the optical map and the luminous flux conservation law into a generalized Monge-Ampère equation. We sketch a least-squares solution strategy.\u3c/p\u3

Numerical dissipation and dispersion of the homogeneous and complete flux schemes

We analyse numerical dissipation and dispersion of the homogeneous flux (HF) and complete flux (CF) schemes, finite volume methods introduced in [4]. To that purpose we derive the modified equation of both schemes . We show that the HF scheme suffers from numerical diffusion for dominant advection, which is effectively removed in the CF scheme. The latter scheme, however, is prone to numerical dispersion. We validate both schemes for a model problem

Compact high order complete flux schemes

\u3cp\u3eIn 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.\u3c/p\u3