G. W. Wei

Discrete singular convolution for the solution of the Fokker–Planck equation

This paper introduces a discrete singular convolution algorithm for solving the Fokker–Planck equation. Singular kernels of the Hilbert-type and the delta type are presented for numerical computations. Various sequences of approximations to the singular kernels are discussed. A numerical algorithm is proposed to incorporate the approximation kernels for physical applications. Three standard problems, the Lorentz Fokker–Planck equation, the bistable model and the Henon–Heiles system, are utilized to test the accuracy, reliability, and speed of convergency of the present approach. All results are in excellent agreement with those of previous methods in the field.

High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources

This paper introduces a novel high order interface scheme, the matched interface and boundary (MIB) method, for solving elliptic equations with discontinuous coefficients and singular sources on Cartesian grids. By appropriate use of auxiliary line and/or fictitious points, physical jump conditions are enforced at the interface. Unlike other existing interface schemes, the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study. To construct higher order interface schemes, the proposed MIB method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions. The proposed MIB method is of arbitrarily high order, in principle. In treating straight, regular interfaces we construct MIB schemes up to 16th-order. For more general elliptic problems with curved, irregular interfaces and boundary, up to 6th-order MIB schemes have been demonstrated. By employing the standard high-order finite difference schemes to discretize the Laplacian, the present MIB method automatically reduces to the standard central difference scheme when the interface is absent. The immersed interface method (IIM) is regenerated for a comparison study of the proposed method. The robustness of the MIB method is verified against the large magnitude of the jump discontinuity across the interface. The nature of high efficiency and low memory requirement of the MIB method is extensively validated via solving various elliptic immersed interface problems in two- and three-dimensions. ee-dimensions.

A NEW BENCHMARK QUALITY SOLUTION FOR THE BUOYANCY-DRIVEN CAVITY BY DISCRETE SINGULAR CONVOLUTION

This article introduces a high-accuracy discrete singular convolution (DSC) for the numerical simulation of coupled convective heat transfer problems. The problem of a buoyancy-driven cavity is solved by two completely independent numerical procedures. One is a quasi-wavelet-based DSC approach, which uses the regularized Shannons kernel, while the other is a standard form of the Galerkin finite-element method. The integration of the Navier-Stokes and energy equations is performed by employing velocity correction-based schemes. The entire laminar natural convection range of 10 3 h Ra h 10 8 is numerically simulated by both schemes. The reliability and robustness of the present DSC approach is extensively tested and validated by means of grid sensitivity and convergence studies. As a result, a set of new benchmark quality data is presented. The study emphasizes quantitative, rather than qualitative comparisons.This article introduces a high-accuracy discrete singular convolution (DSC) for the numerical simulation of coupled convective heat transfer problems. The problem of a buoyancy-driven cavity is solved by two completely independent numerical procedures. One is a quasi-wavelet-based DSC approach, which uses the regularized Shannons kernel, while the other is a standard form of the Galerkin finite-element method. The integration of the Navier-Stokes and energy equations is performed by employing velocity correction-based schemes. The entire laminar natural convection range of 10 3 h Ra h 10 8 is numerically simulated by both schemes. The reliability and robustness of the present DSC approach is extensively tested and validated by means of grid sensitivity and convergence studies. As a result, a set of new benchmark quality data is presented. The study emphasizes quantitative, rather than qualitative comparisons.

A new algorithm for solving some mechanical problems

This paper explores the utility of a discrete singular convolution algorithm for solving certain mechanical problems. Benchmark mechanical systems, including plate vibrations and incompressible flows, are employed to illustrate the robustness and to test accuracy of the present algorithm. Numerical results indicate that the present approach is very accurate, efficient and reliable for solving the aforementioned problems.

Discrete singular convolution for the prediction of high frequency vibration of plates

Theoretical analysis of high frequency vibrations is indispensable in a variety of engineering designs. Much effort has been made on this subject in the past few decades. However, there is no single technique which can be applied with confidence to various engineering structures for high frequency predictions at present. This paper introduces a novel computational approach, the discrete singular convolution (DSC) algorithm, for high frequency vibration analysis of plate structures. Square plates with six distinct boundary conditions are considered. To validate the proposed method, a completely independent approach, the Levy method, is employed to provide exact solutions for a comparison. The proposed method is also validated by convergence studies. We demonstrate the ability of the DSC algorithm for high frequency vibration analysis by providing extremely accurate frequency parameters for plates vibrating in the first 5000 modes.

Generalized Perona-Malik equation for image restoration

This article introduces generalizations of the Perona-Malik (1990) equation. An edge enhancing functional is proposed for direct edge enhancement. A number of super diffusion operators is introduced for fast and effective smoothing. Statistical information is utilized for robust edge-stopping. Numerical integration is conducted by using a previously developed quasi-interpolating wavelet method. Computer experiments indicate that the present algorithm is very efficient for edge-detecting and noise-removing.

The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution

This paper introduces the discrete singular convolution algorithm for vibration analysis of rectangular plates with mixed boundary conditions. A unified scheme is proposed for the treatment of simply supported, clamped and transversely supported (with nonuniform elastic rotational restraint) boundary conditions. The robustness and reliability of the present approach are tested by a number of numerical experiments. All results agree well with those in the literature.

Discrete singular convolution for beam analysis

This paper explores the utility of a discrete singular convolution (DSC) algorithm for beam analysis. Regularized Shannon and Dirichlet kernels are selected to illustrate the present algorithm. Three classes of benchmark beam problems, including bending, vibration and buckling, are utilized to test numerical accuracy and speed of convergence of the present approach. Numerical experiments indicate that the DSC is a simple and reliable algorithm for beam analysis.

Discrete singular convolution for the sine-Gordon equation

Abstract This paper explores the utility of a discrete singular convolution (DSC) algorithm for the integration of the sine-Gordon equation. The initial values are chosen close to a homoclinic manifold for which previous methods have encountered significant numerical difficulties such as numerically induced spatial and temporal chaos. A number of new initial values are considered, including a case where the initial value is “exactly” on the homoclinic orbit. The present algorithm performs extremely well in terms of accuracy, efficiency, simplicity, stability and reliability.

DSC time-domain solution of Maxwell's equations

A new computational algorithm, the discrete singular convolution (DSC), is introduced for solving scattering and guided wave problems described by time-domain Maxwells equations. The DSC algorithm is utilized for the spatial discretization and the fourth-order Runge Kutta scheme is used for the time advancing. Staggered meshes are used for electromagnetic fields. Four standard test problems, a hollow air-filled waveguide, a dielectric slab-loaded rectangular waveguide, a shield microstrip line and a dielectric square, are employed to illustrate the usefulness, to test the accuracy and to explore the limitation of the DSC algorithm. Results are compared with those of finite difference, scaling function multi-resolution time domain, and finite element-based high frequency structure simulator. Numerical experiments indicate that the present algorithm is a promising approach for achieving high accuracy in electromagnetic wave computations.