The weighted continuous galerkin scheme for ordinary differential equations

We introduce the Weighted Continuous Galerkin Scheme for initial value ordinary differential equations. This is an extension of the Continuous Galerkin Scheme, having an extra parameter for the purpose of error reduction. We prove convergence in the L ₂ norm in the time variable in a new way, similar to (elliptic) finite element techniques. Using the optimal L ₂ estimates, we then prove max norm convergence. Numerical evidence for the effectiveness of the proposed scheme is presented.     

A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher's equation

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional time-fractional Fisher's equation, which is obtained from the standard one-dimensional Fisher's equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0<α<1). The proposed LDG is based on the LDG finite element method for space and finite difference method for time. We prove that the method is stable, and the numerical solution converges to the exact one with order O(h^k+1+τ^2?α), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The numerical experiments reveal that the LDG is very effective.     

A Modified Fourier–Galerkin Method for the Poisson and Helmholtz Equations

In this paper we present a modified Fourier–Galerkin method for the numerical solution of the Poisson and Helmholtz equations in a d-dimensional box. The inversion of the differential operators requires O(N ^d ) operations, where N ^d is the number of unknowns. The total cost of the presented algorithms is O(N ^d :log₂:N), due to the application of the Fast Fourier Transform (FFT) at the preprocessing stage. The method is based on an extension of the Fourier spaces by adding appropriate functions. Utilizing suitable bilinear forms, approximate projections onto these extended spaces give rapidly converging and highly accurate series expansions.     

A lumped Galerkin method for solving the Burgers equation

A numerical solution of the one-dimensional Burgers equation is obtained using a lumped Galerkin method with quadratic B-spline finite elements. The scheme is implemented to solve a set of test problems with known exact solutions. Results are compared with published numerical and exact solutions. The proposed scheme performs well. A linear stability analysis shows the scheme to be unconditionally stable.     

Wavelet Galerkin BEM on unstructured meshes

Abstact The present paper is devoted to the fast solution of boundary integral equations on unstructured meshes by the Galerkin scheme. On the given mesh we construct a wavelet basis providing vanishing moments with respect to the traces of polynomials in the space. With this basis at hand, the system matrix in wavelet coordinates can be compressed to O(Nlog N) relevant matrix coefficients, where N denotes the number of unknowns. The compressed system matrix can be computed within suboptimal complexity by using techniques from the fast multipole method or panel clustering. Numerical results prove that we succeeded in developing a fast wavelet Galerkin scheme for solving the considered class of problems. Mathematics Subject Classification (2000) 47A20; 65F50; 65N38; 65R20; 65T60 This work is supported in part by the SFB 393 Numerical Simulation on Massive Parallel Computers funded by the Deutsche Forschungsgemeinschaft. Dedicated to George C. Hsiao on the occasion of his 70th birthday.     

A Discontinuous Galerkin Scheme based on a Space-Time Expansion II. Viscous Flow Equations in Multi Dimensions

In part I of these two papers we introduced for inviscid flow in one space dimension a discontinuous Galerkin scheme of arbitrary order of accuracy in space and time. In the second part we extend the scheme to the compressible Navier-Stokes equations in multi dimensions. It is based on a space-time Taylor expansion at the old time level in which all time or mixed space-time derivatives are replaced by space derivatives using the Cauchy-Kovalevskaya procedure. The surface and volume integrals in the variational formulation are approximated by Gaussian quadrature with the values of the space-time approximate solution. The numerical fluxes at grid cell interfaces are based on the approximate solution of generalized Riemann problems for both, the inviscid and viscous part. The presented scheme has to satisfy a stability restriction similar to all other explicit DG schemes which becomes more restrictive for higher orders. The loss of efficiency, especially in the case of strongly varying sizes of grid cells is circumvented by use of different time steps in different grid cells. The presented time accurate numerical simulations run with local time steps adopted to the local stability restriction in each grid cell. In numerical simulations for the two-dimensional compressible Navier-Stokes equations we show the efficiency and the optimal order of convergence being p+1, when a polynomial approximation of degree p is used.     

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.     

NICE h : a higher-order explicit numerical scheme for integration of constitutive models in plasticity

The article introduces, as a result of further development of the first-order scheme NICE, a simple and efficient higher-order explicit numerical scheme for the integration of a system of ordinary differential equations which is constrained by an algebraic condition (DAE). The scheme is based on the truncated Taylor expansion of the constraint equation with order h of the scheme being determined by the highest exponent in the truncated Taylor series. The integration scheme thus conceived will be named NICE ^h , considering both principal premises of its construction. In conjunction with a direct solution technique used to solve the boundary value problem, the NICE ^h scheme is very convenient for integrating constitutive models in plasticity. The plasticity models are defined mostly by a system of algebraic and differential equations in which the yield criterion represents the constraint condition. To study the properties of the new integration scheme, which, like the forward-Euler scheme, is characterised by its implementation simplicity due to the explicitness of its formulations, a damage constitutive model (Gurson–Tvergaard–Needleman model) is considered. The general opinion that the implicit backward-Euler scheme is much more accurate than the thus-far known explicit schemes is challenged by the introduction of the NICE ^h scheme. The accuracy of the higher-order explicit scheme in the studied cases is significantly higher than the accuracy of the classical backward-Euler scheme, if we compare them under the condition of a similar CPU time consumption.     

A parallel approach of the inverse kinematics solutions for robots using a transputer network

This article presents a parallel method for computing inverse kinematics solutions for robots with closed-form solutions moving along a straight line trajectory specified in Cartesian space. Zhang and Paul's approach¹ is improved for accuracy and speed. Instead of using previous joint positions as proposed by Zhang and Paul, a first order prediction strategy is used to decouple the dependency between joint positions, and a zero order approximation solution is computed. A compensation scheme using Taylor series expansion is applied to obtain the trajectory gradient in joint space to replace the correction scheme proposed by Zhang and Paul. The configuration of a Mitsubishi RV-M1 robot is used for the simulation of a closed-form inverse kinematics solutions. An Alta SuperLink/XL with four transputer nodes is used for parallel implementation. The simulation results show a significant improvement in displacement tracking errors and joint configuration errors along the straight line trajectory. The computational latency is reduced as well. The modified approach proposed in this work is more accurate and faster than Zhang and Paul's approach for robots with closed-form inverse kinematics solutions. © 1996 John Wiley & Sons, Inc.     

Third-order temporal discrete scheme for the non-stationary Navier–Stokes equations

In this article, we suggest a new third-order time discrete scheme for the two-dimensional non-stationary Navier–Stokes equations. After presenting the Galerkin finite element approximation for the spatial discretization, we consider an implicit/explicit time discrete scheme for the problem, which is based on the two-step Adams–Moulton scheme (implicit scheme) for the linear term and the three-step Adams–Bashforth scheme (explicit scheme) for the nonlinear term. In this method, we only need to solve a linearized discrete system at each time step, so the scheme can converge fast and the computational cost can be reduced. Moreover, under some assumptions, we deduce the stability and optimal error estimate for the velocity in L ²-norm.     

A priori error estimates in the finite element method for nonself-adjoint elliptic and parabolic interface problems

Abstract We derive a priori error estimates in the finite element method for nonselfadjoint elliptic and parabolic interface problems in a two-dimensional convex polygonal domain. Optimal H ¹-norm and sub-optimal L ²-norm error estimates are obtained for elliptic interface problems. For parabolic interface problems, the continuous-time Galerkin method is analyzed and an optimal order error estimate in the L ²(0,T;H ¹)-norm is established. Further, a discrete-in-time discontinuous Galerkin method is discussed and a related optimal error estimate is obtained. Keywords: Elliptic and parabolic interface problems, finite element method, spatially discrete scheme, discontinuous Galerkin method, error estimates Mathematics Subject Classification (1991): 65N15, 65N20     

A linear discrete scheme for the Ginzburg–Landau equation

This paper considers a 2D Ginzburg–Landau equation with a periodic initial-value condition. A fully discrete Galerkin–Fourier spectral approximation scheme, which is a linear scheme, is constructed and the dynamical behaviour of the discrete system is then analysed. First, the existence and convergence of global attractors of the discrete system are obtained by a priori estimates and the error estimates of the discrete solution without any restriction on the time step, and the convergence of the discrete scheme is then obtained. The numerical stability of the discrete scheme is proved.     

An A Priori Error Analysis of the hp-Version of the Continuous Galerkin FEM for Nonlinear Initial Value Problems

A continuous Galerkin finite element time-stepping method for the approximation of nonlinear initial value problems is analyzed within an hp-context. We derive a priori error bounds in the L²- and H¹-norm that are explicit with respect to the time steps and the approximation orders. In particular, it is shown that, for analytic solutions (with certain possible start-up singularities) exponential convergence rates can be achieved. Moreover, we prove that the scheme superconverges at the nodal points of the time partition. Numerical experiments illustrate the performance of the method.     

A numerical study of stationary solution of viscous Burgers’ equation using wavelet

In this paper we have studied the numerical stationary solution of viscous Burgers’ equation with Neumann boundary conditions by applying wavelet Galerkin method. Burns et al. [J. Burns, A. Balogh, D.S. Gilliam, and V. I. Shubov, Numerical stationary solutions for a viscous Burgers’ equation, J. Maths. Sys. Est. Contl. 8 (1998), pp. 1–16] have reported that for moderately small viscosity and for certain initial conditions, numerical solution approaches non-constant shock-type stationary solution though only possible actual stationary solution is a constant. We found that the wavelet Galerkin method precisely captures the correct steady-state solution. The solutions obtained were impressive and verify theoretical results.     

Building Blocks for Arbitrary High Order Discontinuous Galerkin Schemes

In this article we propose the use of the ADER methodology of solving generalized Riemann problems to obtain a numerical flux, which is high order accurate in time, for being used in the Discontinuous Galerkin framework for hyperbolic conservation laws. This allows direct integration of the semi-discrete scheme in time and can be done for arbitrary order of accuracy in space and time. The resulting fully discrete scheme in time does not need more memory than an explicit first order Euler time-stepping scheme. This becomes possible because of an extensive use of the governing equations inside the numerical scheme itself via the so-called Cauchy–Kovalewski procedure. We give an efficient algorithm for this procedure for the special case of the nonlinear two-dimensional Euler equations. Numerical convergence results for the nonlinear Euler equations results up to 8th order of accuracy in space and time are shown     

A discontinuous Galerkin solver for Boltzmann–Poisson systems in nano devices

In this paper, we present results of a discontinuous Galerkin (DG) scheme applied to deterministic computations of the transients for the Boltzmann–Poisson system describing electron transport in semiconductor devices. The collisional term models optical-phonon interactions which become dominant under strong energetic conditions corresponding to nano-scale active regions under applied bias. The proposed numerical technique is a finite element method using discontinuous piecewise polynomials as basis functions on unstructured meshes. It is applied to simulate hot electron transport in bulk silicon, in a silicon n⁺–n–n⁺ diode and in a double gated 12 nm MOSFET. Additionally, the obtained results are compared to those of a high order WENO scheme simulation and DSMC (Discrete Simulation Monte Carlo) solvers.     

A Monte Carlo approach for Galerkin radiosity

Galerkin radiosity solves the integral rendering equation by projecting the illumination functions into a set of higher-order basis functions. This paper presents a Monte Carlo approach for Galerkin radiosity to compute the coefficients of the basis functions. The new approach eliminates the problems with edge singularities between adjacent surfaces present in conventional Galerkin radiosity, the time complexity is reduced fromO(K ⁴) toO(K ²) for aK-order basis, and ideally specular energy transport can be simulated. As in conventional Galerkin radiosity, no meshing is required even for large or curved surfaces, thus reducing memory requirements, and no a posteriori Gouraud interpolation is necessary. The new algorithm is simple and can be parallelized on any parallel computer, including massively parallel systems.     

Two-Level Non-Overlapping Schwarz Preconditioners for a Discontinuous Galerkin Approximation of the Biharmonic Equation

We present some two-level non-overlapping additive and multiplicative Schwarz methods for a discontinuous Galerkin method for solving the biharmonic equation. We show that the condition numbers of the preconditioned systems are of the order O( H ³/h ³) for the non-overlapping Schwarz methods, where h and H stand for the fine mesh size and the coarse mesh size, respectively. The analysis requires establishing an interpolation result for Sobolev norms and Poincaré–Friedrichs type inequalities for totally discontinuous piecewise polynomial functions. It also requires showing some approximation properties of the multilevel hierarchy of discontinuous Galerkin finite element spaces.This revised version was published online in July 2005 with corrected volume and issue numbers.     

SYNCHRONIZATION OF CHAOTIC GYRO SYSTEMS USING RECURRENT WAVELET CMAC

This study addresses synchronization of two chaotic gyros by using an adaptive recurrent wavelet cerebellar model articulation controller (RWCMAC). The proposed adaptive RWCMAC system contains an RWCMAC and a robust controller. Based on Lyapunov stability theory, the parameters of RWCMAC are on-line tuned and the robust controller is designed for achieving H ^∞ robust performance. Finally, the proposed adaptive RWCMAC system is applied to synchronize two chaotic gyros. Numerical simulation results demonstrate the effectiveness of the proposed control scheme.     

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.