Homogeneous difference schemes

Тихонов, А. Н.; Samarskii, A.A.

doi:10.1016/0041-5553(62)90005-8

Cited by 104 publications

(50 citation statements)

References 1 publication

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“…To find the approximate eigenvalues of problem (1) with the odd ordinal number λ 2n+1 , we use the technique of exact three-point difference schemes [11,20]. We find the approximate eigenvalues of problem (1) with the even ordinal number λ 2n by integration.…”

Section: Numerical Resultsmentioning

confidence: 99%

Functional-discrete Method for Eigenvalue Transmission Problem with Periodic Boundary Conditions

Макаров

Rossokhata

Bandyrsiĭ³

2005

Comput. Methods Appl. Math.

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-We apply a functional-discrete method with convergence rate not worse than that of a geometric series for an eigenvalue transmission problem with periodic boundary conditions. It is shown that the convergence rate increases with an increase in the ordinal number of trial eigenvalue. Based on asymptotic behavior of eigenvalues of the basic problem and the functional-discrete method, qualitative results concerning the arrangement of the eigenvalues of the original eigenvalue transmission problem with periodic boundary conditions are proved. A number of numerical examples are given to support the theory.

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Section: Numerical Resultsmentioning

confidence: 99%

Functional-discrete Method for Eigenvalue Transmission Problem with Periodic Boundary Conditions

Макаров

Rossokhata

Bandyrsiĭ³

2005

Comput. Methods Appl. Math.

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“…Model error estimates for the stress functions will then quickly lead to sharp consistency error estimates in negative norms, which occur naturally in the error analysis. We remark that negative norm techniques for finite difference methods go back to the work of Tikhonov and Samarskii [33], where they were developed for the construction and analysis of finite difference discretizations of diffusion equations with non-smooth coefficients and on non-uniform meshes.…”

Section: The Atomistic Stress Functionmentioning

confidence: 99%

A priori error analysis of two force-based atomistic/continuum models of a periodic chain

2011

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A priori error analysis of two force-based atomistic/continuum models of a periodic chain. Abstract The force-based quasicontinuum (QCF) approximation is a nonconservative atomistic/continuum hybrid model for the simulation of defects in crystals. We present an a priori error analysis of the QCF method, applied to a one-dimensional periodic chain, that is valid for an arbitrary interaction range, large deformations, and takes coarse-graining into account. Our main tool in this analysis is a new concept of atomistic stress.Moreover, we formulate a new atomistic/continuum coupling mechanism based on coupling stresses instead of forces and extend the a priori analysis to this new method. We show that the new method has several theoretical advantages over the original QCF method.

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“…For the one-dimensional case, he proved that the amplitude error in reflected and transmitted waves is determined by the accuracy by which the jump conditions are discretized, while the phase error is determined by the accuracy of the discretization in the interior of the domain. In the one-dimensional case, Tikhonov and Samarskiȋ's [10] averaging formula was used to obtain a second order accurate approximation without explicitly discretizing the jump conditions. Numerical calculations indicated a significant benefit of combining the second order treatment of the jump conditions with a fourth order method in the interior of the domain.…”

Section: Introductionmentioning

confidence: 99%

An Embedded Boundary Method for the Wave Equation with Discontinuous Coefficients

Kreiss¹,

Petersson²

2006

SIAM J. Sci. Comput.

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A second order accurate embedded boundary method for the two-dimensional wave equation with discontinuous wave propagation speed is described. The wave equation is discretized on a Cartesian grid with constant grid size and the interface (across which the wave speed is discontinuous) is allowed to intersect the mesh in an arbitrary fashion. By using ghost points on either side of the interface, previous embedded boundary techniques for the Neumann and Dirichlet problems are generalized to satisfy the jump conditions across the interface to second order accuracy. The resulting discretization of the jump conditions has the desirable property that each ghost point can be updated independently of all other ghost points, resulting in a fully explicit time-integration method. Numerical examples are given where the method is used to study electro-magnetic scattering of a plane wave by a dielectric cylinder. The numerical solutions are evaluated against the analytical solution due to Mie, and point-wise second order accuracy is confirmed.

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Homogeneous difference schemes

Cited by 104 publications

References 1 publication

Functional-discrete Method for Eigenvalue Transmission Problem with Periodic Boundary Conditions

Functional-discrete Method for Eigenvalue Transmission Problem with Periodic Boundary Conditions

A priori error analysis of two force-based atomistic/continuum models of a periodic chain

An Embedded Boundary Method for the Wave Equation with Discontinuous Coefficients

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