Discrete Calderon’s projections on parallelepipeds and their application to computing exterior magnetic fields for FRC plasmas

Kansa, Edward J.; Shumlak, U.; Tsynkov, Semyon

doi:10.1016/j.jcp.2012.09.033

Cited by 9 publications

(11 citation statements)

References 20 publications

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“…First, we compare the performance of the second-order Difference Potentials Method (DPM) with the second-order Immersed Interface Method (IIM) [14][15][16]. Moreover, we present the result of the fourth-order DPM for the same test problem.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Moreover, interface problems result in non-smooth solutions (or even discontinuous solutions) at the interfaces, and therefore standard numerical methods (finite-difference, finite-element methods, etc.) in any dimension (including 1D) will very often fail to produce accurate approximation of the solutions to the interface problems, and thus special numerical algorithms have to be developed for the approximation of such problems (for instance, see simplified 1D example of interface problem in [8], page 12 and Table 7 on page 14).…”

Section: Introductionmentioning

confidence: 99%

“…Among finite-difference based methods for such problems are the Immersed Boundary Method (IB) ( [24,25], etc. ), the Immersed Interface Method (IIM) ( [14][15][16]1,12], etc. ), the Ghost Fluid Method (GFM) ( [9,17,18,10], etc.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

High-order accurate difference potentials methods for parabolic problems

Albright

Epshteyn

Steffen

2015

Applied Numerical Mathematics

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Highly-accurate numerical methods that can efficiently handle problems with interfaces and/or problems in domains with complex geometry are crucial for the resolution of different temporal and spatial scales in many problems from physics and biology. In this paper we continue the work started in [8], and we use modest one-dimensional parabolic problems as the initial step towards the development of high-order accurate methods based on the Difference Potentials approach. The designed methods are well-suited for variable coefficient parabolic models in heterogeneous media and/or models with nonmatching interfaces and with non-matching grids. Numerical experiments are provided to illustrate high-order accuracy and efficiency of the developed schemes. While the method and analysis are simpler in the one-dimensional settings, they illustrate and test several important ideas and capabilities of the developed approach.

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Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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High-order accurate difference potentials methods for parabolic problems

Albright

Epshteyn

Steffen

2015

Applied Numerical Mathematics

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“…For constant coefficient interior problems, solution by the separation of variables method will remain the most efficient approach in 3D. However, for exterior problems one may use convolution with the discrete fundamental solution (see [26,Appendix C]), accelerated by the fast multipole method. The advantage of this approach is that it automatically takes into account the proper behavior of the solution in the far field.…”

Section: Discussionmentioning

confidence: 99%

A High-Order Numerical Method for the Helmholtz Equation with Nonstandard Boundary Conditions

Britt¹,

Tsynkov²,

Turkel³

2013

SIAM J. Sci. Comput.

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We describe a high-order accurate methodology for the numerical simulation of time-harmonic waves governed by the Helmholtz equation. Our approach combines compact finite difference schemes that provide an inexpensive venue toward high-order accuracy with the method of difference potentials developed by Ryaben'kii. The latter can be interpreted as a generalized discrete version of the method of Calderon's operators in the theory of partial differential equations. The method of difference potentials can accommodate nonconforming boundaries on regular structured grids with no loss of accuracy due to staircasing. It introduces a universal framework for treating boundary conditions of any type. A significant advantage of this method is that changing the boundary condition within a fairly broad variety does not require any major changes to the algorithm and is computationally inexpensive. In this paper, we address various types of boundary conditions using the method of difference potentials. We demonstrate the resulting numerical capabilities by solving a range of nonstandard boundary value problems for the Helmholtz equation. These include problems with variable coefficient Robin boundary conditions (including discontinuous coefficients) and problems with mixed (Dirichlet/Neumann) boundary conditions. In all our simulations, we use a Cartesian grid and a circular boundary curve. For those test cases where the overall solution was smooth, our methodology has consistently demonstrated the design fourth-order rate of grid convergence, whereas when the regularity of the solution was not sufficient, the convergence slowed down, as expected. We also show that every additional boundary condition requires only an incremental additional expense.

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“…However, the auxiliary problem must be solved multiple times in order to construct the boundary system, where the number of solutions needed increases with refinement. Once the boundary system has been obtained, generally two additional solutions of the auxiliary problem are required to obtain the desired solution.The DPM has been applied to the solution of BVPs in fluid dynamics [83], acoustics [84-86] and many other applications [81]. It has usually been applied to regular domains.…”

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confidence: 99%

On the application of the method of difference potentials to linear elastic fracture mechanics

Woodward

Utyuzhnikov

Massin³

2015

Numerical Meth Engineering

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International audienceThe Difference Potential Method (DPM) proved to be a very efficient tool for solving boundary value problems (BVPs) in the case of complex geometries. It allows BVPs to be reduced to a boundary equation without the knowledge of Green's functions. The method has been successfully used for solving very different problems related to the solution of partial differential equations. However, it has mostly been considered in regular (Lipschitz) domains. In the current paper, for the first time, the method has been applied to a problem of linear elastic fracture mechanics. This problem requires solving BVPs in domains containing cracks. For the first time, DPM technology has been combined with the finite element method. Singular enrichment functions, such as those used within the extended finite element formulations, are introduced into the system in order to improve the approximation of the crack tip singularity. Near-optimal convergence rates are achieved with the application of these enrichment functions. For the DPM, the reduction of the BVP to a boundary equation is based on generalised surface projections. The projection is fully determined by the clear trace. In the current paper, for the first time, the minimal clear trace for such problems has been numerically realised for a domain with a cut. KEY WORDS: method of difference potentials; extended finite element method; fracture; rate of convergenc

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Discrete Calderon’s projections on parallelepipeds and their application to computing exterior magnetic fields for FRC plasmas

Cited by 9 publications

References 20 publications

High-order accurate difference potentials methods for parabolic problems

High-order accurate difference potentials methods for parabolic problems

A High-Order Numerical Method for the Helmholtz Equation with Nonstandard Boundary Conditions

On the application of the method of difference potentials to linear elastic fracture mechanics

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