A high-order compact scheme for the one-dimensional Helmholtz equation with a discontinuous coefficient

In this paper, based on the idea of the immersed interface method, a fourth-order compact finite difference scheme is proposed for solving one-dimensional Helmholtz equation with discontinuous coefficient, jump conditions are given at the interface. The Dirichlet boundary condition and the Neumann boundary condition are considered. The Neumann boundary condition is treated with a fourth-order scheme. Numerical experiments are included to confirm the accuracy and efficiency of the proposed method.     

High-order compact scheme for boundary points

In this paper, we introduce a new high-order scheme for boundary points when calculating the derivative of smooth functions by compact scheme. The primitive function reconstruction method of ENO schemes is applied to obtain the conservative form of the compact scheme. Equations for approximating the derivatives around the boundary points 1 and N are determined. For the Neumann (and mixed) boundary conditions, high-order equations are derived to determine the values of the function at the boundary points, 1 and N, before the primitive function reconstruction method is applied. We construct a subroutine that can be used with Dirichlet, Neumann, or mixed boundary conditions. Numerical tests are presented to demonstrate the capabilities of this new scheme, and a comparison to the lower-order boundary scheme shows its advantages.     

Finite Difference Method for the Black–Scholes Equation Without Boundary Conditions

We present an accurate and efficient finite difference method for solving the Black–Scholes (BS) equation without boundary conditions. The BS equation is a backward parabolic partial differential equation for financial option pricing and hedging. When we solve the BS equation numerically, we typically need an artificial far-field boundary condition such as the Dirichlet, Neumann, linearity, or partial differential equation boundary condition. However, in this paper, we propose an explicit finite difference scheme which does not use a far-field boundary condition to solve the BS equation numerically. The main idea of the proposed method is that we reduce one or two computational grid points and only compute the updated numerical solution on that new grid points at each time step. By using this approach, we do not need a boundary condition. This procedure works because option pricing and computation of the Greeks use the values at a couple of grid points neighboring an interesting spot. To demonstrate the efficiency and accuracy of the new algorithm, we perform the numerical experiments such as pricing and computation of the Greeks of the vanilla call, cash-or-nothing, power, and powered options. The computational results show excellent agreement with analytical solutions.     

A new fast multipole boundary element method for two dimensional acoustic problems

A new fast multipole boundary element method (BEM) is presented in this paper for solving large-scale two dimensional (2D) acoustic problems based on the improved Burton–Miller formulation. This algorithm has several important improvements. The fast multipole BEM employs the improved Burton–Miller formulation, and successfully overcomes the non-uniqueness difficulty associated with the conventional BEM for exterior acoustic problems. The improved Burton–Miller formulation contains only weakly singular integrals, and avoids the numerical difficulties associated to the evaluation of the hypersingular integral, it leads to the numerical implementations more efficient and straightforward. Furthermore, the fast multipole method (FMM) and the approximate inverse preconditioned generalized minimum residual method (GMRES) iterative solver are adopted to greatly improve the overall computational efficiency. The numerical examples with Neumann boundary conditions are presented that clearly demonstrate the accuracy and efficiency of the developed fast multipole BEM for solving large-scale 2D acoustic problems in a wide range of frequencies.     

Radial Slit Maps of Bounded Multiply Connected Regions

In this paper we present a boundary integral equation method for the numerical conformal mapping of a bounded multiply connected region onto a radial slit region. The method is based on some uniquely solvable boundary integral equations with adjoint classical, adjoint generalized and modified Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to illustrate the efficiency of the presented method.     

A novel method for modeling Neumann and Robin boundary conditions in smoothed particle hydrodynamics

We present a novel smoothed particle hydrodynamics (SPH) method for diffusion equations subject to Neumann and Robin boundary conditions. The Neumann and Robin boundary conditions are common to many physical problems (such as heat/mass transfer), and can prove challenging to implement in numerical methods when the boundary geometry is complex. The new method presented here is based on the approximation of the sharp boundary with a diffuse interface and allows an efficient implementation of the Neumann and Robin boundary conditions in the SPH method. The paper discusses the details of the method and the criteria for the width of the diffuse interface. The method is used to simulate diffusion and reactions in a domain bounded by two concentric circles and reactive flow between two parallel plates and its accuracy is demonstrated through comparison with analytical and finite difference solutions. To further illustrate the capabilities of the model, a reactive flow in a porous medium was simulated and good convergence properties of the model are demonstrated.     

An energy-preserving algorithm for nonlinear Hamiltonian wave equations with Neumann boundary conditions

In this paper, a novel energy-preserving numerical scheme for nonlinear Hamiltonian wave equations with Neumann boundary conditions is proposed and analyzed based on the blend of spatial discretization by finite element method (FEM) and time discretization by Average Vector Field (AVF) approach. We first use the finite element discretization in space, which leads to a system of Hamiltonian ODEs whose Hamiltonian can be thought of as the semi-discrete energy of the original continuous system. The stability of the semi-discrete finite element scheme is analyzed. We then apply the AVF approach to the Hamiltonian ODEs to yield a new and efficient fully discrete scheme, which can preserve exactly (machine precision) the semi-discrete energy. The blend of FEM and AVF approach derives a new and efficient numerical scheme for nonlinear Hamiltonian wave equations. The numerical results on a single-soliton problem and a sine-Gordon equation are presented to demonstrate the remarkable energy-preserving property of the proposed numerical scheme.     

Development of boundary conditions for direct numerical simulations of three-dimensional vortex breakdown phenomena in semi-infinite domains

The suitability of various open boundary conditions is evaluated for direct numerical simulations of three-dimensional, incompressible, spatially and temporally evolving, swirling laminar jets in domains that extend to infinity in the downstream and radial direction. From the point of view of specifying conditions at the open boundaries, this class of flows is particularly challenging due to its ability to support traveling waves. Towards this end, several radial boundary conditions are implemented and tested with respect to their ability to conserve local and global mass, to handle low and high entrainment flow, and to avoid the introduction of artificial waves propagating from the boundaries into the interior: a free-slip condition, two types of homogeneous Neumann conditions, and a radiation condition in spirit of the outflow boundary condition. Global mass is conserved automatically within machine accuracy in the free-slip and simple radiation case, while the Neumann conditions require some iterative modification to conserve mass. This yields a computationally less efficient scheme which additionally exhibits poorer conservation properties due to the limited number of iterations. The free-slip condition typically requires the largest radial extent of the computational domain due to its impermeable character which is particularly problematic for the high entrainment flow. Hence, the radiation condition has been found as the most suitable lateral boundary condition for both high and low entrainment jets.     

Solving multidimensional knapsack problems with generalized upper bound constraints using critical event tabu search

A critical event tabu search method which navigates both sides of the feasibility boundary has been shown effective for solving the multidimensional knapsack problem. In this paper, we apply the method to the multidimensional knapsack problem with generalized upper bound constraints. This paper also demonstrates the merits of using surrogate constraint information vs. a Lagrangian relaxation scheme as choice rules for the problem class. A constraint normalization method is presented to strengthen the surrogate constraint information and improve the computational results. The advantages of intensifying the search at critical solutions are also demonstrated.     

An iterative method for solving 2D wave problems in infinite domains

This paper presents a new method for solving two-dimensional wave problems in infinite domains. The method yields a solution that satisfies Sommerfeld's radiation condition, as required for the correct solution of infinite domains excited only locally. It is obtained by iterations. An infinite domain is first truncated by introducing an artificial finite boundary (β), on which some boundary conditions are imposed. The finite computational domain in each iteration is subjected to actual boundary conditions and to different (Dirichlet or Neumann) fictive boundary conditions on β.     

A new fourth-order difference scheme for solving an N-carrier system with Neumann boundary conditions

In this paper, a numerical method is developed to solve an N-carrier system with Neumann boundary conditions. First, we apply the compact finite difference scheme of fourth order for discretizing spatial derivatives at the interior points. Then, we develop a new combined compact finite difference scheme for the boundary, which also has fourth-order accuracy. Lastly, by using a Padé approximation method for the resulting linear system of ordinary differential equations, a new compact finite difference scheme is obtained. The present scheme has second-order accuracy in time direction and fourth-order accuracy in space direction. It is shown that the scheme is unconditionally stable. The present scheme is tested by two numerical examples, which show that the convergence rate with respect to the spatial variable from the new scheme is higher and the solution is much more accurate when compared with those obtained by using other previous methods.     

An efficient ADI lambda formulation

This paper is concerned with the numerical simulation of compressible inviscid flows, by means of an accurate and efficient technique. The “implicit lambda scheme,” recently presented by the authors for the cases of quasi-one-dimensional flows and two-dimensional flows past thin airfoils, is generalized here to arbitrary two-dimensional geometries. Starting from the time-dependent Euler equations in vector form, simplified by assuming homentropic flow conditions, the lambda-formulation equations are derived for a general orthogonal coordinate system, linearized in time and solved by an alternating direction implicit method. Such a technique is very accurate, due to its use of characteristic-type variables and of upwind differences—which correctly take into account the direction of wave propagation—and efficient, insofar as it allows to overcome the CFL stability limitation of explicit methods, while requiring the solution of only block-tridiagonal systems. The importance of the choice of the computational grid and the boundary conditions on the accuracy and, henceforth, on the efficiency of the calculations is analyzed in some detail. The marits of the present approach are demonstrated by means of a few applications.     

An improved CE/SE scheme for multi-material elastic-plastic flows and its applications

In this paper, a new definition of SE and CE, which is based on the hexahedron mesh and simpler than Chang’s original CE/SE method (the space-time Conservation Element and Solution Element method), is proposed and an improved CE/SE scheme is constructed. Furthermore, the improved CE/SE scheme is extended in order to solve the elastic-plastic flow problems. The hybrid particle level set method is used for tracing the interfaces of materials. Proper boundary conditions are presented in interface tracking. Two high-velocity impact problems are simulated numerically and the computational results are carefully compared with the experimental data, as well as the results from other literature and LS-DYNA software. The comparisons show that the computational scheme developed currently is clear in physical concept, easy to be implemented and high accurate and efficient for the problems considered.     

Numerical consistency and spurious boundary layer in the projection method

In this work, the controversial issue regarding the boundary condition for the Laplace equation has been addressed. It turned out to be that the velocity field can be superimposed by a gradient of the Helmholtz potential part that satisfies the Neumann type boundary condition and a correction term. By this representation it follows that the physical variables (u,P) are independent of the flux imposed along the computational boundary at each fractional step. Provided that, the same flux distribution is imposed at each time step along the computational boundary in the decoupled case. Furthermore, for each fractional component, the Helmholtz term generates pseudo-boundary layers, in opposite sign, along the computational boundary, to accommodate the imposed flux. The summation of both component annihilates the pseudo-boundary layer. A failure to comply with the restriction that the same flux distribution should be imposed at each time step may result in inconsistency of the solution. For a flux that is slowly varying over time, this type of inconsistency may lower the order of the scheme accuracy, while for a rapid variation over time the solution is inconsistent. The rapid variation in time is related to the initial stage of the decoupled problem with a given flux along the computational boundary. For such a case an educated guess is required for the initial velocity-potential, to prevent an inconsistent solution at an early stage of the solution.     

A Newton method for Bernoulli’s free boundary problem in three dimensions

Summary The present paper is dedicated to the numerical solution of Bernoulli’s free boundary problem in three dimensions. We reformulate the given free boundary problem as a shape optimization problem and compute the shape gradient and Hessian of the given shape functional. To approximate the shape problem we apply a Ritz–Galerkin discretization. The necessary optimality condition is resolved by Newton’s method. All information of the state equation, required for the optimization algorithm, are derived by boundary integral equations which we solve numerically by a fast wavelet Galerkin scheme. Numerical results confirm that the proposed Newton method yields an efficient algorithm to treat the considered class of problems.      

Reversible data hiding with context modeling,generalized expansion and boundary map

This paper proposes a reversible data hiding scheme with high capacity-distortion efficiency, which embeds data by expanding prediction-errors. Instead of using the MED predictor as did in other schemes, a predictor with context modeling, which refines prediction-errors through an error feedback mechanism, is adopted to work out prediction-errors. The context modeling can significantly sharpen the distribution of prediction-errors, and benefit the embedding capacity and the image quality. To expand prediction-errors, the proposed scheme utilizes a generalized expansion, which enables it to provide capacities larger than 1 bpp (bits per pixel) without resorting to multiple embedding. Besides, a novel boundary map is proposed to record overflow-potential pixels. The boundary map is much shorter compared with either a location map or an overflow map even though it is not compressed. The combination of the context modeling, the generalized expansion and the boundary map makes the overall scheme efficient in pursuing large embedding capacity and high image quality. Experimental results demonstrate that the proposed scheme provides competitive capacity compared with other state-of-the-art schemes when the image quality is kept at the same level.     

The boundary element-free method for 2D interior and exterior Helmholtz problems

The boundary element-free method (BEFM) is developed in this paper for numerical solutions of 2D interior and exterior Helmholtz problems with mixed boundary conditions of Dirichlet and Neumann types. A unified boundary integral equation is established for both interior and exterior problems. By using the improved interpolating moving least squares method to form meshless shape functions, mixed boundary conditions in the BEFM can be satisfied directly and easily. Detailed computational formulas are derived to compute weakly and strongly singular integrals over linear and higher order integration cells. Three numerical integration procedures are developed for the computation of strongly singular integrals. Numerical examples involving acoustic scattering and radiation problems are presented to show the accuracy and efficiency of the meshless method.     

Flow field computation for the high voltage gas blast circuit breaker with the moving boundary

We studied the gas dynamics for the ideal gas in the simplified high voltage (HV) gas blast circuit breaker with the moving boundary. The piston and the electric contact are moving. Since the boundary is moving, it is difficult for the ordinary finite difference (FD) method or the finite element (FE) method to compute the solution. For the purpose of numerical simplicity and efficiency, we introduced an upwind meshfree scheme which is an excellent scheme for the time varying domain. Despite the low coding and computational cost, the numerical simulation is successfully conducted. Our method is even more efficient when considering a three-dimensional computation with a moving boundary.     

Partial semi-coarsening multigrid method based on the HOC scheme on nonuniform grids for the convection–diffusion problems

A partial semi-coarsening multigrid method based on the high-order compact (HOC) difference scheme on nonuniform grids is developed to solve the 2D convection–diffusion problems with boundary or internal layers. The significance of this study is that the multigrid method allows different number of grid points along different coordinate directions on nonuniform grids. Numerical experiments on some convection–diffusion problems with boundary or internal layers are conducted. They demonstrate that the partial semi-coarsening multigrid method combined with the HOC scheme on nonuniform grids, without losing the high-order accuracy, is very efficient and effective to decrease the computational cost by reducing the number of grid points along the direction which does not contain boundary or internal layers.     

A general scheme for log-determinant computation of matrices via stochastic polynomial approximation

We study the approximation of determinant for large scale matrices with low computational complexity. This paper develops a generalized stochastic polynomial approximation frame as well as a stochastic Legendre approximation algorithm to calculate log-determinants of large-scale positive definite matrices based on the prior eigenvalue distributions. The generalized frame is implemented by weighted $L_2$ orthogonal polynomial expansions with an efficient recursion formula and matrix–vector multiplications. So the proposed scheme is efficient both in computational complexity and data storage. Respective error bounds are given in theory which guarantee the convergence of the proposed algorithms. We illustrate the effectiveness of our method by numerical experiments on both synthetic matrices and counting spanning trees.