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A generalized boundary integral equation method for boundary value problems of two-dimensional isotropic lattice Laplacian is proposed in this paper. The proposed method is an extension of the classical boundary integral equation method with notable advantage. By utilizing the asymptotic expression of the fundamental solution at infinity, this method effectively addresses the challenge of numerical integration involving singular integral kernels. The introduction of Green’s formulas, Dirichlet and Neumann traces, and other tools which are parallel to the traditional integral equation method, form a solid foundation for the development of the generalized boundary integral equation method. The solvability of boundary integral equations and the solvability of lattice interface problem are important guarantees for the feasibility of this method, and these are emphasized in this paper. Subsequently, the generalized boundary integral equation method is applied to boundary value problems equipped with either Dirichlet or Neumann boundary conditions. Simple numerical examples demonstrate the accuracy and effectiveness of the generalized boundary integral equation method. [ABSTRACT FROM AUTHOR]

Time constant equilibration reduction (TICER) is a local model reduction method based on Gaussian elimination for RC networks which was introduced in Sheehan (1999 IEEE/ACM international conference on computer-aided design. Digest of technical papers (Cat. No.99CH37051), pp 200–203, 1999). The main idea of TICER is selectively removing the non-port nodes of the circuits to produce a smaller RC circuit. Since proposed, TICER has been widely used in the electronic engineering, and lots of practices validate its reliability and robustness. It is a practical method to meet the needs of reduction and can produce a realizable RC circuit with good accuracy, and this is the reason why TICER has been used until now. Later, many scholars proposed RC-realizable reduction methods based on TICER. Quite surprisingly, as far as the authors know, there exists no rigorous analysis on this simple but wonderful method in the literature. In this short communication, we prove the optimal first-order asymptotic accuracy of TICER and present numerical evidences to validate our theoretical results. [ABSTRACT FROM AUTHOR]

A space reduction method for structural operator equations is proposed in this paper. It turns out that many interface problems derived by applying the idea of domain decomposition can be categorized into this framework. A seemingly simple algebraic technique is proposed to reduce the complexity of operator equations. The connection between this technique and the integral equation method is revealed. Under mild conditions, we prove that the reduced operator equation by space reduction is well-posed, and its solution is the same as that of the original one. As two applications, we apply the proposed method to solve a planar triangular lattice problem and an exterior problem of modified Helmholtz equation with FEM discretization. The numerical evidence validates the effectiveness. [ABSTRACT FROM AUTHOR]

For the Cauchy problem of the high frequency wave-type equation with Wentzel–Kramers–Brillouin (WKB) type initial data, the extended Wentzel–Kramers–Brillouin (E-WKB) ansatz is an asymptotically valid solution. This ansatz is globally defined and formulated as an integral of coherent states over the displaced Lagrangian submanifold. This paper proves the optimal first order error estimate of the proposed E-WKB ansatz in L 2 norm for the wave-type equation in the semi-classical regime. The key ingredients in the proof are the moving frame technique developed in Zheng (Commun Math Sci 11:105–140, 2013) and the deep relations between the E-WKB analysis and the classical WKB analysis. Numerical results on the linear KdV equation verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

We extend the theory of global geometrical optics method, proposed originally for the linear scalar high-frequency wave-like equations in [Commun. Math. Sci., 2013, 11(1): 105-140], to the more general vector-valued Schrödinger problems in the semi-classical regime. The key ingredient in the global geometrical optics method is a moving frame technique in the phase space. The governing equation is transformed into a new equation but of the same type when expressed in any moving frame induced by the underlying Hamiltonian flow. The classical Wentzel-Kramers-Brillouin (WKB) analysis benefits from this treatment as it maintains valid for arbitrary but fixed evolutionary time. It turns out that a WKB-type function defined merely on the underlying Lagrangian submanifold can be obtained with the help of this moving frame technique, and from which a uniform first-order approximation of the wave field can be derived, even around caustics. The general theory is exemplified by two specific instances. One is the two-level Schrödinger system and the other is the periodic Schrödinger equation. Numerical tests validate the theoretical results. [ABSTRACT FROM AUTHOR]

Fast evaluation of the exact transparent boundary condition for the one-dimensional cubic nonlinear Schrödinger equation is considered in this paper. In [J. Comput. Math., 2007, 25(6): 730–745], the author proposed a fast evaluation method for the half-order time derivative operator. In this paper, we apply this method for the exact transparent boundary condition for the one-dimensional cubic nonlinear Schrödinger equation. Numerical tests demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The numerical simulation of the solution to a modified KdV equation on the whole real axis is considered in this paper. Based on the work of Fokas (Comm Pure Appl Math 58(5):639–670, 2005), a kind of exact nonreflecting boundary conditions which are suitable for numerical purposes are presented with the inverse scattering theory. With these boundary conditions imposed on the artificially introduced boundary points, a reduced problem defined on a finite computational interval is formulated. The discretization of the nonreflecting boundary conditions is studied in detail, and a dual-Petrov–Galerkin spectral method is proposed for the numerical solution to the reduced problem. Some numerical tests are given, which validate the effectiveness, and suggest the stability of the proposed scheme. [ABSTRACT FROM AUTHOR]