33 Result(s)
within
J. H. M. ten Thije Boonkkamp
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Chapter and Conference Paper
An Energy-Preserving High Order Method for Liouville’s Equation of Geometrical Optics
Liouville’s equation describes light propagation through an optical system. It governs the evolution of an energy distribution on phase space. This distribution is discontinuous across optical interfaces. The ...
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Article
Open AccessNumerical methods for the hyperbolic Monge-Ampère equation based on the method of characteristics
We present three alternative derivations of the method of characteristics (MOC) for a second order nonlinear hyperbolic partial differential equation (PDE) in two independent variables. The MOC gives rise to t...
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Article
Correction to: An Energy Conservative hp-method for Liouville’s Equation of Geometrical Optics
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Article
Open AccessAn Energy Conservative hp-method for Liouville’s Equation of Geometrical Optics
Liouville’s equation on phase space in geometrical optics describes the evolution of an energy distribution through an optical system, which is discontinuous across optical interfaces. The discontinuous Galerk...
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Chapter and Conference Paper
Generalized Monge–Ampère Equations for Freeform Optical System Design
We present the derivation of the generalized Monge–Ampère equation for two optical systems, viz. a freeform lens with parallel incident and refracted light rays, which transforms a source emittance into a desi...
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Chapter and Conference Paper
Novel Flux Approximation Schemes for Systems of Coupled Advection-Diffusion-Reaction Equations
The physical modeling of transport in multi-component mixtures results in systems of coupled equations for the mass fractions. This contribution discusses the mathematical structure of such transport systems a...
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Article
Open AccessA Monge–Ampère Problem with Non-quadratic Cost Function to Compute Freeform Lens Surfaces
In this article, we present a least-squares method to compute freeform surfaces of a lens with parallel incoming and outgoing light rays, which is a transport problem corresponding to a non-quadratic cost functio...
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Chapter and Conference Paper
Double Freeform Lens Design for Laser Beam Sha**: A Least-Squares Approach
The location of the surfaces of a double freeform lens, required for laser beam sha**, is governed by a Monge-Ampère type equation. We outline a least-squares solver and demonstrate the performance of the me...
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Chapter and Conference Paper
Nonlinear Flux Approximation Scheme for Burgers Equation Derived from a Local BVP
We present a novel flux approximation scheme for the viscous Burgers equation. The numerical flux is computed from a local two-point boundary value problem for the stationary equation and requires the iterativ...
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Chapter and Conference Paper
A Least-Squares Method for a Monge-Ampère Equation with Non-quadratic Cost Function Applied to Optical Design
Freeform optical surfaces can transfer a given light distribution of the source into a desired distribution at the target. Freeform optical design problems can be formulated as a Monge-Ampère type differential...
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Chapter and Conference Paper
A Nonlinear Flux Approximation Scheme for the Viscous Burgers Equation
We a nonlinear flux approximation for the discretization of the Burgers equation. We derive the numerical flux function from a local two-point boundary value problem (BVP), which results in a nonlinear...
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Chapter and Conference Paper
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 ...
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Chapter and Conference Paper
Complete Flux Scheme for Conservation Laws Containing a Linear Source
We present an extension of the complete flux scheme for conservation laws containing a linear source. In our new scheme, we split off the linear part of the source and incorporate this term in the homogeneous ...
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Chapter and Conference Paper
Flux Approximation Scheme for the Incompressible Navier-Stokes Equations Using Local Boundary Value Problems
We present a flux approximation scheme for the incompressible Navier-Stokes equations, that is based on a flux approximation scheme for the scalar advection-diffusion-reaction equation that we developed earlie...
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Chapter and Conference Paper
Discretization and Parallel Iterative Schemes for Advection-Diffusion-Reaction Problems
Conservation laws of advection-diffusion-reaction (ADR) type are ubiquitous in continuum physics. In this paper we outline discretization of these problems and iterative schemes for the resulting linear system...
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Chapter and Conference Paper
A Sub-cell Discretization Method for the Convective Terms in the Incompressible Navier-Stokes Equations
In this contribution we present a sub-cell discretization method for the computation of the interface velocities involved in the convective terms of the incompressible Navier-Stokes equations. We compute an in...
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Chapter and Conference Paper
Harmonic Complete Flux Schemes for Conservation Laws with Discontinuous Coefficients
In this paper we discuss several complete flux schemes for advection-diffusion-reaction problems. We consider both scalar equations as well as systems of equations. For the flux approximations in the latter ca...
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Chapter and Conference Paper
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 scheme...
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Chapter and Conference Paper
The Complete Flux Scheme in Cylindrical Coordinates
We consider the complete flux (CF) scheme , a finite volume method (FVM) presented in [3]. CF is based on an integral representation for the fluxes, found by solving a local boundary value problem that includes t...
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Chapter and Conference Paper
A New Discretization Method for the Convective Terms in the Incompressible Navier-Stokes Equations
In this contribution we present the use of local one-dimensional boundary value problems (BVPs) to compute the interface velocities in the convective terms of the incompressible Navier-Stokes equations. This t...
