A Legendre Pseudospectral Penalty Scheme for Solving Time-Domain Maxwell’s Equations

In this paper we present a Legendre pseudospectral algorithm based on a tensor product formulation for solving the time-domain Maxwell equations. Our approach starts by conducting an analysis for finding well-posed boundary operators for the Maxwell equations. We then discuss equivalent characteristic boundary conditions for common physical boundary constraints. These theoretical results are then employed to construct a pseudospectral penalty scheme which is asymptotically stable at the semidiscrete level. Numerical computations based on the proposed scheme are also provided for different cases where exact solutions exist. By measuring the differences between the computed and exact solutions, we observe the expected convergence patterns of the scheme. This work is supported by National Science Council grant No. NSC 95-2120-M-001-003.     

Polynomial time-marching for nonperiodic boundary value problems

A polynomial interpolation time-marching technique can efficiently provide balanced spectral accuracy in both the space and time dimensions for some PDEs. The Newton-form interpolation based on Fejér points has been successfully implemented to march the periodic Fourier pseudospectral solution in time. In this paper, this spectrally accurate time-stepping technique will be extended to solve some typical nonperiodic initial boundary value problems by the Chebyshev collocation spatial approximation. Both homogeneous Neumann and Dirichlet boundary conditions will be incorporated into the time-marching scheme. For the second order wave equation, besides more accurate timemarching, the new scheme numerically has anO(1/N ²) time step size limitation of stability, much larger thanO(1/N ⁴) stability limitation in conventional finitedifference time-stepping, Chebyshev space collocation methods.     

Numerical analysis of a new conservative scheme for the coupled nonlinear Schrödinger equations

We devote the present paper to an efficient conservative scheme for the coupled nonlinear Schrödinger (CNLS) system, based on the Fourier pseudospectral method, the Crank–Nicolson method and leap-frog method. To obtain the present scheme, the key idea consists of two aspects. First, we solve the CNLS system based on its Hamiltonian structure and the resulted scheme can preserve the Hamiltonian nature. Second, we use Fourier pseudospectral method in spatial discretization and Crank–Nicolson/ leap-frog scheme for discretizing linear/ nonlinear terms in time direction, respectively. The proposed scheme is energy-preserving, mass-preserving, uniquely solvable and unconditionally stable, while being decoupled, linearized and suitable for parallel computation in practical computation. Using the energy method and the classical interpolation theory, an error estimate of the proposed scheme is proven strictly without any grid ratio restrictions in the discrete L² norm. Finally, numerical results are reported to verify our theoretical analysis.     

Stable Boundary Treatment for the Wave Equation on Second-Order Form

A stable and accurate boundary treatment is derived for the second-order wave equation. The domain is discretized using narrow-diagonal summation by parts operators and the boundary conditions are imposed using a penalty method, leading to fully explicit time integration. This discretization yields a stable and efficient scheme. The analysis is verified by numerical simulations in one-dimension using high-order finite difference discretizations, and in three-dimensions using an unstructured finite volume discretization.     

Adaptive patch-based mesh fitting for reverse engineering

In this paper,  we propose a novel adaptive mesh fitting algorithm that fits a triangular model with G¹ smoothly stitching bi-quintic Bézier patches. Our algorithm first segments the input mesh into a set of quadrilateral patches, whose boundaries form a quadrangle mesh. For each boundary of each quadrilateral patch, we construct a normal curve and a boundary-fitting curve, which fit the normal and position of its boundary vertices respectively. By interpolating the normal and boundary-fitting curves of each quadrilateral patch with a Bézier patch, an initial G¹ smoothly stitching Bézier patches is generated. We perform this patch-based fitting scheme in an adaptive fashion by recursively subdividing the underlying quadrilateral into four sub-patches. The experimental results show that our algorithm achieves precision-ensured Bézier patches with G¹ continuity and meets the requirements of reverse engineering.     

A Fourth Order Accurate Finite Difference Scheme for the Elastic Wave Equation in Second Order Formulation

We present a fourth order accurate finite difference method for the elastic wave equation in second order formulation, where the fourth order accuracy holds in both space and time. The key ingredient of the method is a boundary modified fourth order accurate discretization of the second derivative with variable coefficient, (??(x)u _x ) _x . This discretization satisfies a summation by parts identity that guarantees stability of the scheme. The boundary conditions are enforced through ghost points, thereby avoiding projections or penalty terms, which often are used with previous summation by parts operators. The temporal discretization is obtained by an explicit modified equation method. Numerical examples with free surface boundary conditions show that the scheme is stable for CFL-numbers up to 1.3, and demonstrate a significant improvement in efficiency over the second order accurate method. The new discretization of (??(x)u _x ) _x has general applicability, and will enable stable fourth order accurate approximations of other partial differential equations as well as the elastic wave equation.     

A Hybrid Fourier–Chebyshev Method for Partial Differential Equations

We propose a pseudospectral hybrid algorithm to approximate the solution of partial differential equations (PDEs) with non-periodic boundary conditions. Most of the approximations are computed using Fourier expansions that can be efficiently obtained by fast Fourier transforms. To avoid the Gibbs phenomenon, super-Gaussian window functions are used in physical space. Near the boundaries, we use local polynomial approximations to correct the solution. We analyze the accuracy and eigenvalue stability of the method for several PDEs. The method compares favorably to traditional spectral methods, and numerical results indicate that for hyperbolic problems a time step restriction of O(1/N) is sufficient for stability. R.B. Platte’s address after December 2009: Arizona State University, Department of Mathematics and Statistics, Tempe, AZ, 85287-1804.     

快速收敛的四边形网格三分细分模式

提出了四边形网格的三分细分模式.对于正则和非正则四边形网格,分别采用不同的细分模板获得新的细分顶点.从双三次B样条中推导出正则四边形网格的三分细分模板,极限曲面C²连续;对细分矩阵进行傅里叶变换,推导出非正则四边形网格的三分细分模板,极限曲面C¹连续.提出的三分细分模式可以解决任意拓扑四边形网格的曲面细分问题.与其他细分模式相比,具有收敛速度快、适用范围广等优点.最后给出了四边形网格细分的实例.     

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.     

Efficient removal of boundary-divergence errors in time-splitting methods

A normal mode analysis is presented and numerical tests are performed to assess the effectiveness of a new time-splitting algorithm proposed recently in Karniadakiset al. (1990) for solving the incompressible Navier-Stokes equations. This new algorithm employs high-order explicit pressure boundary conditions and mixed explicit/implicit stiffly stable time-integration schemes, which can lead to arbitrarily high-order accuracy in time. In the current article we investigate both the time accuracy of the new scheme as well as the corresponding reduction in boundary-divergence errors for two model flow problems involving solid boundaries. The main finding is that time discretization errors, induced by the nondivergent splitting mode, scale with the order of the accuracy of the integration rule employed if a proper rotational form of the pressure boundary condition is used; otherwise a first-order accuracy in time similar to the classical splitting methods is achieved. In the former case the corresponding errors in divergence can be completely eliminated, while in the latter case they scale asO(vt)_1/2.     

Accuracy of the Simultaneous-Approximation-Term Boundary Condition for Time-Dependent Problems

The simultaneous-approximation-term (SAT) approach to applying boundary conditions for the compressible Navier-Stokes equations is analyzed with respect to the errors associated with the formulation’s weak enforcement of the boundary data. Three numerical examples are presented which illustrate the relationship between the penalty parameters and the accuracy; two examples are fundamentally acoustic and the third is viscous. The viscous problem is further analyzed by a continuous model whose solution is known analytically and which approximates the discrete problem. From the analysis it is found that at early times an overshoot in the boundary values relative to the boundary data can be expected for all values of the penalty parameters but whose amplitude reduces with the inverse of the parameter. Likewise, the long-time behavior exhibits a t ^−1/2 relaxation towards the specified data, but with a very small amplitude. Based on these data it is evident that large values of the penalty parameters are not required for accuracies comparable to those obtained by a more traditional characteristics-based method. It is further found that for curvilinear boundaries the SAT approach is superior to the locally one-dimensional inviscid characteristic approach.     

Quad/Triangle Subdivision

In this paper we introduce a new subdivision operator that unifies triangular and quadrilateral subdivision schemes. Designers often want the added flexibility of having both quads and triangles in their models. It is also well known that triangle meshes generate poor limit surfaces when using a quad scheme, while quad‐only meshes behave poorly with triangular schemes. Our new scheme is a generalization of the well known Catmull‐Clark and Loop subdivision algorithms. We show that our surfaces are C ¹ everywhere and provide a proof that it is impossible to construct such a C ² scheme at the quad/triangle boundary. However, we provide rules that produce surfaces with bounded curvature at the regular quad/triangle boundary and provide optimal masks that minimize the curvature divergence elsewhere. We demonstrate the visual quality of our surfaces with several examples. ACM CSS: I.3.5 Computer Graphics—Curve, surface, solid, and object representations     

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.     

Discontinuous Galerkin Method for Linear Free-Surface Gravity Waves

In this paper, we discuss a discontinuous Galerkin finite (DG) element method for linear free surface gravity waves. We prove that the algorithm is unconditionally stable and does not require additional smoothing or artificial viscosity terms in the free surface boundary condition to prevent numerical instabilities on a non-uniform mesh. A detailed error analysis of the full time-dependent algorithm is given, showing that the error in the wave height and velocity potential in the L²-norm is in both cases of optimal order and proportional to O(Δt²+h^p+1), without the need for a separate velocity reconstruction, with p the polynomial order, h the mesh size and Δt the time step. The error analysis is confirmed with numerical simulations. In addition, a Fourier analysis of the fully discrete scheme is conducted which shows the dependence of the frequency error and wave dissipation on the time step and mesh size. The algebraic equations for the DG discretization are derived in a way suitable for an unstructured mesh and result in a symmetric positive definite linear system. The algorithm is demonstrated on a number of model problems, including a wave maker, for discretizations with accuracy ranging from second to fourth order.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Absorbing Boundary Conditions of the Second Order for the Pseudospectral Chebyshev Methods for Wave Propagation

In this paper we describe the implementation of one-way wave equations of the second order in conjuction with pseudospectral methods for wave propagation in two space dimensions. These equations are first reformulated as hyperbolic systems of the first order and the absorbing boundaries are implemented by an appropriate modification of the matrix of this system. The resulting matrix corresponding to one-way wave equation based on Padé approximation has all eigenvalues in the complex negative half plane which allows stable integration of the underlying system by any ODE solver in the sense of eigenvalue stability. The obtained numerical scheme is much more accurate than the schemes obtained before which utilized absorbing boundary conditions of the first order, and is also capable of integrating the wave propagation problems on much larger time intervals than was previously possible.     

Finite difference method for time-space-fractional Schrödinger equation

In this paper, an implicit finite difference scheme for the nonlinear time-space-fractional Schrödinger equation is presented. It is shown that the implicit scheme is unconditionally stable with experimental convergence order of O(τ^2?α+h²), where τ and h are time and space stepsizes, respectively, and α (0<α<1) is the fractional-order in time. In order to reduce the computational cost, the explicit–implicit scheme is proposed such that the nonlinear term is easily treated. Meanwhile, the implicit finite difference scheme for the coupled time-space-fractional Schrödinger system is also presented, which is unconditionally stable too. Numerical examples are given to support the theoretical analysis.     

Co-volume method for Riemannian mean curvature flow in subjective surfaces multiscale segmentation

We introduce semi-implicit complementary volume numerical scheme for solving the level setformulation of Riemannian mean curvature flow problem arising in image segmentation, edge detection, missing boundary completion and subjective contour extraction. The scheme is robust and efficient since it is linear, and it is stable in L_∞ and weighted W ^1,1 sense without any restriction on a time step. The computational results related to medical image segmentation with partly missing boundaries and subjective contours extraction are presented.     

A New Interpolatory Subdivision for Quadrilateral Meshes

This paper presents a new interpolatory subdivision scheme for quadrilateral meshes based on a 1–4 splitting operator. The scheme generates surfaces coincident with those of the Kobbelt interpolatory subdivision scheme for regular meshes. A new group of rules are designed for computing newly inserted vertices around extraordinary vertices. As an extension of the regular masks,the new rules are derived based on a reinterpretation of the regular masks. Eigen‐structure analysis demonstrates that subdivision surfaces generated using the new scheme are C¹ continuous and, in addition, have bounded curvature.     

Bounded Error Schemes for the Wave Equation on Complex Domains

This paper considers the application of the method of boundary penalty terms (SAT) to the numerical solution of the wave equation on complex shapes with Dirichlet boundary conditions. A theory is developed, in a semi-discrete setting, that allows the use of a Cartesian grid on complex geometries, yet maintains the order of accuracy with only a linear temporal error-bound. A numerical example, involving the solution of Maxwell’s equations inside a 2-D circular wave-guide demonstrates the efficacy of this method in comparison to others (e.g., the staggered Yee scheme)—we achieve a decrease of two orders of magnitude in the level of the L₂-error.     

Compact difference method for solving the fractional reaction–subdiffusion equation with Neumann boundary value condition

In this paper, we derive a high-order compact finite difference scheme for solving the reaction–subdiffusion equation with Neumann boundary value condition. The L1 method is used to approximate the temporal Caputo derivative, and the compact difference operator is applied for spatial discretization. We prove that the compact finite difference method is unconditionally stable and convergent with order O(τ^2?α+h⁴) in L₂ norm, where τ, α, and h are the temporal step size, the order of time fractional derivative and the spatial step size, respectively. Finally, some numerical experiments are carried out to show the effectiveness of the proposed difference scheme.