Partial Differential Equations

Mattheij, R.M.M.; Rienstra, S.W.; Boonkkamp, J. H. M. ten Thije

doi:10.1137/1.9780898718270

Cited by 116 publications

(73 citation statements)

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“…One very useful tool in the analysis of hyperbolic PDEs is the method of characteristics (MOC) [42]. The MOC effectively turns a hyperbolic PDE into a set of ordinary differential equations.…”

Section: Methods Of Characteristicsmentioning

confidence: 99%

A Novel Scheme for Liouville’s Equation with a Discontinuous Hamiltonian and Applications to Geometrical Optics

et al. 2016

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A novel scheme is developed that computes numerical solutions of Liouville's equation with a discontinuous Hamiltonian. It is assumed that the underlying Hamiltonian system has well-defined behaviour even when the Hamiltonian is discontinuous. In the case of geometrical optics such a discontinuity yields the familiar Snell's law or the law of specular reflection. Solutions to Liouville's equation should be constant along curves defined by the Hamiltonian system when the right-hand side is zero, i.e., no absorption or collisions. This consideration allows us to derive a new jump condition, enabling us to construct a first-order accurate scheme. Essentially, the correct physics is built into the solver. The scheme is tested in a two-dimensional optical setting with two test cases, the first using a single jump in the refractive index and the second a compound parabolic concentrator. For these two situations, the scheme outperforms the more conventional method of Monte Carlo ray tracing.

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Section: Methods Of Characteristicsmentioning

confidence: 99%

A Novel Scheme for Liouville’s Equation with a Discontinuous Hamiltonian and Applications to Geometrical Optics

et al. 2016

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“…If sgn(α) = sgn(β), then a harmonic function satisfying the Robin boundary condition (1) is unique 15 .…”

Section: Theoremmentioning

confidence: 99%

Solving a class of Robin problems in simply connected regions via integral equations with a generalized Neumann kernel

Al-Shatri¹,

Murid²,

Ismail³

2017

ScienceAsia

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This paper presents a new boundary integral equation method for the solution of a class of Robin problems in bounded and unbounded simply connected regions. We show how to reformulate the Robin problems as RiemannHilbert problems which lead to systems of integral equations, related differential equations, and normalizing conditions that give rise to unique solutions. Numerical results on several test regions are presented to illustrate the solution technique for the Robin problems when the boundaries are sufficiently smooth.

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“…Several techniques exist to study the relation between the local and global error, see for example [11,7,15]. We use the approach outlined in Wesseling [27].…”

Section: Global Truncation Errormentioning

confidence: 99%

Boundary treatment for fourth-order staggered mesh discretizations of the incompressible Navier–Stokes equations

Sanderse

Verstappen

Koren

2014

Journal of Computational Physics

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Harlow and Welch [Phys. Fluids 8 (1965) 2182-2189] introduced a discretization method for the incompressible Navier-Stokes equations conserving the secondary quantities kinetic energy and vorticity, besides the primary quantities mass and momentum. This method was extended to fourth order accuracy by several researchers [25,14,21]. In this paper we propose a new consistent boundary treatment for this method, which is such that continuous integration-by-parts identities (including boundary contributions) are mimicked in a discrete sense. In this way kinetic energy is exactly conserved even in case of nonzero tangential boundary conditions. We show that requiring energy conservation at the boundary conflicts with order of accuracy conditions, and that the global accuracy of the fourth order method is limited to second order in the presence of boundaries. We indicate how non-uniform grids can be employed to obtain full fourth order accuracy.

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Partial Differential Equations

Cited by 116 publications

References 0 publications

A Novel Scheme for Liouville’s Equation with a Discontinuous Hamiltonian and Applications to Geometrical Optics

A Novel Scheme for Liouville’s Equation with a Discontinuous Hamiltonian and Applications to Geometrical Optics

Solving a class of Robin problems in simply connected regions via integral equations with a generalized Neumann kernel

Boundary treatment for fourth-order staggered mesh discretizations of the incompressible Navier–Stokes equations

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