378 research outputs found

    On the stability of a weighted finite difference scheme for wave equation with nonlocal boundary conditions

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    We consider the stability of a weighted finite difference scheme for a linear hyperbolic equation with nonlocal integral boundary condition. By studying the spectrum of the transition matrix of the three-layered difference scheme we obtain a sufficient stability condition in a special matrix norm. *The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014)

    A survey on stationary problems, Green's functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions

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    In this paper, we present a survey of recent results on the Green's functions and on spectrum for stationary problems with nonlocal boundary conditions. Results of Lithuanian mathematicians in the field of differential and numerical problems with nonlocal boundary conditions are described. *The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014)

    Finite Difference Method for the Reverse Parabolic Problem

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    A finite difference method for the approximate solution of the reverse multidimensional parabolic differential equation with a multipoint boundary condition and Dirichlet condition is applied. Stability, almost coercive stability, and coercive stability estimates for the solution of the first and second orders of accuracy difference schemes are obtained. The theoretical statements are supported by the numerical example

    A Note on the Second Order of Accuracy Stable Difference Schemes for the Nonlocal Boundary Value Hyperbolic Problem

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    The second order of accuracy absolutely stable difference schemes are presented for the nonlocal boundary value hyperbolic problem for the differential equations in a Hilbert space H with the self-adjoint positive definite operator A. The stability estimates for the solutions of these difference schemes are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions and multidimensional hyperbolic equation with Dirichlet conditions are considered. The stability estimates for the solutions of these difference schemes for the nonlocal boundary value hyperbolic problem are established. Finally, a numerical method proposed and numerical experiments, analysis of the errors, and related execution times are presented in order to verify theoretical statements

    On Stability of a Third Order of Accuracy Difference Scheme for Hyperbolic Nonlocal BVP with Self-Adjoint Operator

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    A third order of accuracy absolutely stable difference schemes is presented for nonlocal boundary value hyperbolic problem of the differential equations in a Hilbert space H with self-adjoint positive definite operator A. Stability estimates for solution of the difference scheme are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions is considered

    Numerical solution of nonlinear elliptic equation with nonlocal condition

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    Two iterative methods are considered for the system of difference equations approximating two-dimensional nonlinear elliptic equation with the nonlocal integral condition. Motivation and possible applications of the problem present in the paper coincide with the small volume problems in hydrodynamics. The differential problem considered in the article is some generalization of the boundary value problem for minimal surface equation

    Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting)

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    This small collaborative workshop brought together experts from the Sino-German project working in the field of advanced numerical methods for hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the convergence of numerical methods and proper solution concepts were addressed as well
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