Numerical meshless solution of high-dimensional sine-Gordon equations via Fourier HDMR-HC approximation

In this paper, an implicit time stepping meshless scheme is proposed to find the numerical solution of high-dimensional sine-Gordon equations (SGEs) by combining the high dimensional model representation (HDMR) and the Fourier hyperbolic cross (HC) approximation. To ensure the sparseness of the relevant coefficient matrices of the implicit time stepping scheme, the whole domain is first divided into a set of subdomains, and the relevant derivatives in high-dimension can be separately approximated by the Fourier HDMR-HC approximation in each subdomain. The proposed method allows for stable large time-steps and a relatively small number of nodes with satisfactory accuracy. The numerical examples show that the proposed method is very attractive for simulating the high-dimensional SGEs

Numerical Simulation of Two Dimentional sine-Gordon Solitons Using the Modified Cubic B-Spline Differential Quadrature Method

In this article, a numerical simulation of two dimensional nonlinear sine-Gordon equation with Neumann boundary condition is obtained by using a composite scheme referred to as a modified cubic B spline differential quadrature method. The modified cubic B-spline serves as a basis function in the differential quadrature method to compute the weighting coefficients. Thus, the sine-Gordon equation is converted into a system of second order ordinary differential equations (ODEs). We solve the resulting system of ODEs by an optimal five stage and fourth-order strong stability preserving Runge Kutta scheme. Both damped and undamped cases are considered for the numerical simulation with Josephson current density function with value minus one. The computed results are found to be in good agreement with the exact solutions and other numerical results available in literature

An Efficient Finite Difference Scheme for the 2D Sine-Gordon Equation

We present an efficient second-order finite difference scheme for solving the 2D sine-Gordon equation, which can inherit the discrete energy conservation for the undamped model theoretically. Due to the semi-implicit treatment for the nonlinear term, it leads to a sequence of nonlinear coupled equations. We use a linear iteration algorithm, which can solve them efficiently, and the contraction mapping property is also proven. Based on truncation errors of the numerical scheme, the convergence analysis in the discrete $l^2$-norm is investigated in detail. Moreover, we carry out various numerical simulations, such as verifications of the second order accuracy, tests of energy conservation and circular ring solitons, to demonstrate the efficiency and the robustness of the proposed scheme.Comment: 18 pages, 7 figure

Efficient energy-preserving numerical approximations for the sine-Gordon equation with Neumann boundary conditions

We present two novel classes of fully discrete energy-preserving algorithms for the sine-Gordon equation subject to Neumann boundary conditions. The cosine pseudo-spectral method is first used to develop structure-preserving spatial discretizations under two different meshes, which result two finite-dimensional Hamiltonian ODE systems. Then we combine the prediction-correction Crank-Nicolson scheme with the projection approach to arrive at fully discrete energy-preserving methods. Alternatively, we introduce a supplementary variable to transform the initial model into a relaxation system, which allows us to construct structure-preserving algorithms more easily. We then discretize the relaxation system directly by using the cosine pseudo-spectral method in space and the prediction-correction Crank-Nicolson scheme in time to derive a new class of energy-preserving schemes. The proposed methods can be solved effectively by the discrete Cosine transform. Some benchmark examples and numerical comparisons are presented to demonstrate the accuracy, efficiency and superiority of the proposed schemes

Virtual element method for semilinear sine-Gordon equation over polygonal mesh using product approximation technique

In this paper, we employ the linear virtual element spaces to discretize the semilinear sine-Gordon equation in two dimensions. The salient features of the virtual element method (VEM) are: (a) it does not require explicit form of the shape functions to construct the nonlinear and the bilinear terms, and (b) relaxes the constraint on the mesh topology by allowing the domain to be discretized with general polygons consisting of both convex and concave elements, and (c) easy mesh refinements (hanging nodes and interfaces are allowed). The nonlinear source term is discretized by employing the product approximation technique and for temporal discretization, the Crank-Nicolson scheme is used. The resulting nonlinear equations are solved using the Newton's method. We derive a priori error estimations in $L^2$ and $H^1$ norms. The convergence properties and the accuracy of the virtual element method for the solution of the sine-Gordon equation are demonstrated with academic numerical experiments.Comment: 28 pages, 3 Figures, 3 Table

On the simulation of the energy transmission in the forbidden band-gap of a spatially discrete double sine-Gordon system

In this work, we present a numerical method to consistently approximate solutions of a spatially discrete, double sine-Gordon chain which considers the presence of external damping. In addition to the finite-difference scheme employed to approximate the solution of the difference-differential equations of the model under investigation, our method provides positivity-preserving schemes to approximate the local and the total energy of the system, in such way that the discrete rate of change of the total energy with respect to time provides a consistent approximation of the corresponding continuous rate of change. Simulations are performed, first of all, to assess the validity of the computational technique against known qualitative solutions of coupled sine-Gordon and coupled double sine-Gordon chains. Secondly, the method is used in the investigation of the phenomenon of nonlinear transmission of energy in double sine-Gordon systems; the qualitative effects of the damping coefficient on the occurrence of the nonlinear process of supratransmission are briefly determined in this work, too

A linearly implicit and local energy-preserving scheme for the sine-Gordon equation based on the invariant energy quadratization approach

In this paper, we develop a novel, linearly implicit and local energy-preserving scheme for the sine-Gordon equation. The basic idea is from the invariant energy quadratization approach to construct energy stable schemes for gradient systems, which are energy dispassion. We here take the sine-Gordon equation as an example to show that the invariant energy quadratization approach is also an efficient way to construct linearly implicit and local energy-conserving schemes for energy-conserving systems. Utilizing the invariant energy quadratization approach, the sine-Gordon equation is first reformulated into an equivalent system, which inherits a modified local energy conservation law. The new system are then discretized by the conventional finite difference method and a semi-discretized system is obtained, which can conserve the semi-discretized local energy conservation law. Subsequently, the linearly implicit structure-preserving method is applied for the resulting semi-discrete system to arrive at a fully discretized scheme. We prove that the resulting scheme can exactly preserve the discrete local energy conservation law. Moveover, with the aid of the classical energy method, an unconditional and optimal error estimate for the scheme is established in discrete $H_h^1$-norm. Finally, various numerical examples are addressed to confirm our theoretical analysis and demonstrate the advantage of the new scheme over some existing local structure-preserving schemes.Comment: 26 page

Local weak form meshless techniques based on the radial point interpolation (RPI) method and local boundary integral equation (LBIE) method to evaluate European and American options

For the first time in mathematical finance field, we propose the local weak form meshless methods for option pricing; especially in this paper we select and analysis two schemes of them named local boundary integral equation method (LBIE) based on moving least squares approximation (MLS) and local radial point interpolation (LRPI) based on Wu's compactly supported radial basis functions (WCS-RBFs). LBIE and LRPI schemes are the truly meshless methods, because, a traditional non-overlapping, continuous mesh is not required, either for the construction of the shape functions, or for the integration of the local sub-domains. In this work, the American option which is a free boundary problem, is reduced to a problem with fixed boundary using a Richardson extrapolation technique. Then the $\theta$-weighted scheme is employed for the time derivative. Stability analysis of the methods is analyzed and performed by the matrix method. In fact, based on an analysis carried out in the present paper, the methods are unconditionally stable for implicit Euler (\theta = 0) and Crank-Nicolson (\theta = 0.5) schemes. It should be noted that LBIE and LRPI schemes lead to banded and sparse system matrices. Therefore, we use a powerful iterative algorithm named the Bi-conjugate gradient stabilized method (BCGSTAB) to get rid of this system. Numerical experiments are presented showing that the LBIE and LRPI approaches are extremely accurate and fast

Fundamental concepts and models for the direct problem

This book series is an initiative of the Post Graduate Program in Integrity of Engineering Materials from UnB, organized as a collaborative work involving researchers, engineers, scholars, from several institutions, universities, industry, recognized both nationally and internationally. The book chapters discuss several direct methods, inverse methods and uncertainty models available for model-based and signal based inverse problems, including discrete numerical methods for continuum mechanics (Finite Element Method, Boundary Element Method, Mesh-Free Method, Wavelet Method). The different topics covered include aspects related to multiscale modeling, multiphysics modeling, inverse methods (Optimization, Identification, Artificial Intelligence and Data Science), Uncertainty Modeling (Probabilistic Methods, Uncertainty Quantification, Risk & Reliability), Model Validation and Verification. Each book includes an initial chapter with a presentation of the book chapters included in the volume, and their connection and relationship with regard to the whole setting of methods and models