A sparse grid space-time discretization scheme for parabolic problems

Summary In this paper, we consider the discretization in space and time of parabolic differential equations where we use the so-called space-time sparse grid technique. It employs the tensor product of a one-dimensional multilevel basis in time and a proper multilevel basis in space. This way, the additional order of complexity of a direct space-time discretization can be avoided, provided that the solution fulfills a certain smoothness assumption in space-time, namely that its mixed space-time derivatives are bounded. This holds in many applications due to the smoothing properties of the propagator of the parabolic PDE (heat kernel). In the more general case, the space-time sparse grid approach can be employed together with adaptive refinement in space and time and then leads to similar approximation rates as the non-adaptive method for smooth functions. We analyze the properties of different space-time sparse grid discretizations for parabolic differential equations from both, the theoretical and practical point of view, discuss their implementational aspects and report on the results of numerical experiments.      

Closed-form approximation of the periodic response of a system with a saturating non-linearity†

A technique for approximating harmonic, periodic solutions of a class of non-linear equations involving u saturation non-linearity is applied to a fourth-order equation which describes an electrical circuit containing a non-linear amplifying element. The computational effort required to obtain the closed-form approximation which this method yields is compared with that required for a numerical solution by the conventional methods of quasilinearization and patching. The various methods use comparable amounts of computer time, but the other classical methods do not offer the advantage of the closed-form approximation which our technique provides.     

Piecewise quasilinearization techniques for singular boundary-value problems

Piecewise quasilinearization methods for singular boundary-value problems in second-order ordinary differential equations are presented. These methods result in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus yielding piecewise analytical solutions. The accuracy of the globally smooth piecewise quasilinear method is assessed by comparisons with exact solutions of several Lane-Emden equations, a singular problem of non-Newtonian fluid dynamics and the Thomas-Fermi equation. It is shown that the smooth piecewise quasilinearization method provides accurate solutions even near the singularity and is more precise than (iterative) second-order accurate finite difference discretizations. It is also shown that the accuracy of the smooth piecewise quasilinear method depends on the kind of singularity, nonlinearity and inhomogeneities of singular ordinary differential equations. For the Thomas-Fermi equation, it is shown that the piecewise quasilinearization method that provides globally smooth solutions is more accurate than that which only insures global continuity, and more accurate than global quasilinearization techniques which do not employ local linearization.     

A Hybrid Implicit-Explicit Adaptive Multirate Numerical Scheme for Time-Dependent Equations

We develop a hybrid implicit and explicit adaptive multirate time integration method to solve systems of time-dependent equations that present two significantly different scales. We adopt an iteration scheme to decouple the equations with different time scales. At each iteration, we use an implicit Galerkin method with a fast time-step to solve for the fast scale variables and an explicit method with a slow time-step to solve for the slow variables. We derive an error estimator using a posteriori analysis which controls both the iteration number and the adaptive time-step selection. We present several numerical examples demonstrating the efficiency of our scheme and conclude with a stability analysis for a model problem.     

The large discretization step method for time-dependent partial differential equations

A new method for the acceleration of linear and nonlinear time-dependent calculations is presented. It is based on the large discretization step (LDS, in short) approximation, defined in this work, which employs an extended system of low accuracy schemes to approximate a high accuracy discrete approximation to a time-dependent differential operator.These approximations are efficiently implemented in the LDS methods for linear and nonlinear hyperbolic equations, presented here. In these algorithms the high and low accuracy schemes are interpreted as the same discretization of a time-dependent operator on fine and coarse grids, respectively. Thus, a system of correction terms and corresponding equations are derived and solved on the coarse grid to yield the fine grid accuracy. These terms are initialized by visiting the fine grid once in many coarse grid time steps. The resulting methods are very general, simple to implement and may be used to accelerate many existing time marching schemes.The efficiency of the LDS algorithms is defined as the cost of computing the fine grid solution relative to the cost of obtaining the same accuracy with the LDS methods. The LDS method’s typical efficiency is 16 for two-dimensional problems and 28 for three-dimensional problems for both linear and nonlinear equations. For a particularly good discretization of a linear equation, an efficiency of 25 in two-dimensional and 66 in three-dimensional was obtained.     

Computational methods for the tubular chemical reactor

The numerical methods that have been successfully applied to the tubular reactor are here reviewed. After demonstrating the essential stiffness of the equations, the Newton-Raphson iteration and collocation methods are described for the steady state. For the calculation of transients the collocation methods can be usefully supplemented by quasilinearization.     

A high order ADI method for separable generalized Helmholtz equations

We present a multilevel high order ADI method for separable generalized Helmholtz equations. The discretization method we use is a one-dimensional fourth order compact finite difference applied to each directional component of the Laplace operator, resulting in a discrete system efficiently solvable by ADI methods. We apply this high order difference scheme to all levels of grids, and then starting from the coarsest grid, solve the discretized equation with an ADI method at each grid level, with the solution from the previous grid level as the initial guess. The multilevel procedure stops as the ADI finishes its iterations on the finest grid. Analytical and experimental results show that the proposed method is highly accurate and efficient while remaining as algorithmically and data-structurally simple as the single grid ADI method.     

Adaptive finite-volume solution of complex turbulent flows

This paper describes an automatic local grid adaptation procedure driven by an evaluation of the differential residuals of the RANS equations computed using a higher-order reconstruction operator. A suitable data structure is developed for the local mesh adaptation process to be flexible and low CPU time consuming. The whole procedure is designed in the framework of finite-volume methods on unstructured grids. To avoid the appearance of ill-conditioned near-wall cells in the vicinity of curved surfaces of bodies a global mesh deformation technique is used. The whole procedure is applied to a complex turbulent flow around a high-lift multiple element airfoil in take-off configuration using the Spalart-Allmaras turbulence model. The adaptation is controlled by as many indicators as there are equations involved in the problem. It is demonstrated that the proposed methodology performs rigorous local adaptive mesh refinement and automatically achieves grid independent results. Thus, interesting gains are obtained in terms of CPU time, memory requirement and user effort compared to single mesh computations.     

On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems

For the singular, non-Hermitian, and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the semi-convergence of the Hermitian and skew-Hermitian splitting (HSS) iteration methods. We then investigate the semi-convergence factor and estimate its upper bound for the HSS iteration method. If the semi-convergence condition is satisfied, it is shown that the semi-convergence rate is the same as that of the HSS iteration method applied to a linear system with the coefficient matrix equal to the compression of the original matrix on the range space of its Hermitian part, that is, the matrix obtained from the original matrix by restricting the domain and projecting the range space to the range space of the Hermitian part. In particular, an upper bound is obtained in terms of the largest and the smallest nonzero eigenvalues of the Hermitian part of the coefficient matrix. In addition, applications of the HSS iteration method as a preconditioner for Krylov subspace methods such as GMRES are investigated in detail, and several examples are used to illustrate the theoretical results and examine the numerical effectiveness of the HSS iteration method served either as a preconditioner for GMRES or as a solver.     

基于保辛算法的绳系卫星系统闭环反馈控制问题求解

应用非线性系统滚动时域控制的保辛算法求解绳系卫星系统子星释放和回收过程的闭环反馈控制问题.通过第二类Lagrange方程推导出二体绳系卫星系统的动力学方程;通过拟线性化方法将绳系卫星系统闭环反馈控制问题转化为线性非齐次Hamilton系统两端边值问题的迭代求解;通过保辛算法将线性非齐次Hamilton两端边值问题转化为线性方程组的求解;通过递进更新时间步的状态变量和控制变量,完成绳系卫星系统的闭环反馈控制.数值仿真表明:相对于Legendre伪谱方法,用保辛算法求解绳系卫星系统的闭环反馈控制问题的计算速度和收敛速度较快.绳系卫星系统的开环控制和闭环反馈控制问题数值仿真结果表明:在绳系卫星的初始状态存在偏差的情况下,使用开环控制会导致系统在终端无法达到稳定状态,而使用闭环反馈控制则能在一段时间内抵消初始状态向量偏差对系统产生的影响,最终达到稳定状态.     

Numerical solution of reactive-diffusive systems

Four time linearization techniques and two operator-splitting algorithms have been employed to study the propagation of a one-dimensional wave governed by a reaction-diffusion equation. Comparisons amongst the methods are shown in terms of the L ²-norm error and computed wave speeds. The calculations have been performed with different numerical grids in order to determine the effects of the temporal and spatial step sizes on the accuracy. It is shown that a time linearization procedure with a second-order accurate temporal approximation and a fourth-order accurate spatial discretization yields the most accurate results. The numerical calculations are compared with those reported in Parts 1 and 2. It is concluded that the most accurate time linearization method described in this paper offers a great promise for the computation of multi-dimensional reaction-diffusion equations.     

Numerical solution of incompressible Navier–Stokes equations using a fractional-step approach

A fractional step method for the solution of steady and unsteady incompressible Navier–Stokes equations is outlined. The method is based on a finite-volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (third and fifth) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when fifth-order upwind differencing and a modified production term in the Baldwin–Barth one-equation turbulence model are used with adequate grid resolution.     

Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media

In this paper, we study the robust linearization of nonlinear poromechanics of unsaturated materials. The model of interest couples the Richards equation with linear elasticity equations, generalizing the classical Biot equations. In practice a monolithic solver is not always available, defining the requirement for a linearization scheme to allow the use of separate simulators. It is not met by the classical Newton method. We propose three different linearization schemes incorporating the fixed-stress splitting scheme, coupled with an L-scheme, Modified Picard and Newton linearization of the flow equations. All schemes allow the efficient and robust decoupling of mechanics and flow equations. In particular, the simplest scheme, the Fixed-Stress-L-scheme, employs solely constant diagonal stabilization, has low cost per iteration, and is very robust. Under mild, physical assumptions, it is theoretically shown to be a contraction. Due to possible break-down or slow convergence of all considered splitting schemes, Anderson acceleration is applied as post-processing. Based on a special case, we justify theoretically the general ability of the Anderson acceleration to effectively accelerate convergence and stabilize the underlying scheme, allowing even non-contractive fixed-point iterations to converge. To our knowledge, this is the first theoretical indication of this kind. Theoretical findings are confirmed by numerical results. In particular, Anderson acceleration has been demonstrated to be very effective for the considered Picard-type methods. Finally, the Fixed-Stress-Newton scheme combined with Anderson acceleration shows the best performance among the splitting schemes.     

Error function scheme for Burgers' equation

The present paper is devoted to the development of a new scheme to solve the one-dimensional time-dependent Burgers' equation locally on sub-domains, using similarity reductions for partial differential equations. Each sub-domain is divided into three grid points. The ordinary differential equation deduced from the similarity reduction can be integrated and is then used to approximate the flux vector in the Burgers' equation. The arbitrary constants in the analytical solution of the similarity equation can be determined in terms of the dependent variables at the grid points in each sub-domain. This approach eliminates the difficulties associated with boundary conditions for the similarity reductions over the whole solution domain. Numerical results are obtained for two different test cases and are compared with other numerical results.     

Two-parameter generalized Hermitian and skew-Hermitian splitting iteration method

It is known that the Hermitian and skew-Hermitian splitting (HSS) iteration method is an efficient solver for non-Hermitian positive-definite linear system of equations. Benzi [A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl. 31 (2009), pp. 360–374] proposed a generalized HSS (GHSS) iteration method. In this paper, we present a two-parameter version of the GHSS (TGHSS) method and investigate its convergence properties. To show the effectiveness of the proposed method the TGHSS iteration method is applied to image restoration and convection–diffusion problems and the results are compared with those of the HSS and GHSS methods.     

Solution of a nonlinear time-delay model in biology via semi-analytical approaches

The delay logistic equations have been extensively used as models in biology and other sciences, with particular emphasis on population dynamics. In this work, the variational iteration and Adomian decomposition methods are applied to solve the delay logistic equation. The variational iteration method is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. On the other hand, the Adomian decomposition method approximates the solution as an infinite series and usually converges to the accurate solution. Moreover, these techniques reduce the volume of calculations because they have no need of discretization of the variables, linearization or small perturbations. Illustrative examples are included to demonstrate the validity and applicability of the presented methods.     

Fluid formulation of high intensity laser beam propagation using lagrangian coordinates

The complete mathematical modeling of nonlinear light-matter interaction is presented in a hydrodynamic context. The field intensity and the phase gradient are the dependent variables of interest. The resulting governing equations are a generalization of the Navier-Stokes equations. This fluid formulation allows the insights and the methodologies which have been gained in solving hydrodynamics problems to be extended to nonlinear optics problems. To insure effective numerical treatment of the anticipated nonlinear self-lensing phenomena, a self-adjusted nonuniform redistribution, along the direction of propagation, of the computation points according to the actual local requirements of the physics must be used. As an alternative to the application of adaptive rezoning techniques in conjunction with Eulerian coordinates, Lagrangian variables are used to provide automatically the desired nonlinear mapping from the physical plane into the mathematical frame. In this paper we propose a method suitable for the solution of the described problemin one-dimensional cases as well as in two- dimensional cases with cylindrical symmetry. To overcome the numerical difficulties related to the inversion of the Jacobian, an analytical algorithm based on the paraxial approximation was developed.     

Preconditioned HSS iteration method and its non-alternating variant for continuous Sylvester equations

Bai (2010) proposed an efficient Hermitian and skew-Hermitian splitting (HSS) iteration method for solving a broad class of large sparse continuous Sylvester equations. To further improve the efficiency of the HSS method, in this paper we present a preconditioned HSS (PHSS) iteration method and its non-alternating variant (NPHSS) for this matrix equation. The convergence properties of the PHSS and NPHSS methods are studied in depth and the quasi-optimal values of the iteration parameters for the two methods are also derived. Moreover, to reduce the computational cost, we establish the inexact variants of the two iteration methods. Numerical experiments illustrate the efficiency and robustness of the two iteration methods and their inexact variants.     

Piecewise-linearized and linearized θ-methods for ordinary and partial differential equations

Initial- and boundary-value problems appear frequently in many branches of physics. In this paper, several numerical methods, based on linearization techniques, for solving these problems are reviewed. First, piecewise-linearized methods and linearized θ-methods are considered for the solution of initial-value problems in ordinary differential equations. Second, piecewise-linearized techniques for two-point boundary-value problems in ordinary differential equations are developed and used in conjunction with a shooting method. In order to overcome the lack of convergence associated with shooting, piecewise-linearized methods which provide piecewise analytical solutions and yield nonstandard finite difference schemes are presented. Third, methods of lines in either space or time for the solution of one-dimensional convection-reaction-diffusion problems that transform the original problem into an initial- or boundary-value one are reviewed. Methods of lines in time that result in boundary-value problems at each time step can be solved by means of the techniques described here, whereas methods of lines in space that yield initial-value problems and employ either piecewise-linearized techniques or linearized θ-methods in time are also developed. Finally, for multidimensional problems, approximate factorization methods are first used to transform the multidimensional problem into a sequence of one-dimensional ones which are then solved by means of the linearized and piecewise-linearized methods presented here.     

A composite numerical scheme for the numerical simulation of coupled Burgers’ equation

In this work, a composite numerical scheme based on finite difference and Haar wavelets is proposed to solve time dependent coupled Burgers’ equation with appropriate initial and boundary conditions. Time derivative is discretized by forward difference and then quasilinearization technique is used to linearize the coupled Burgers’ equation. Space derivatives discretization with Haar wavelets leads to a system of linear equations and is solved using Matlab7.0. Convergence analysis of proposed scheme exhibits that the error bound is inversely proportional to the resolution level of the Haar wavelet. Finally, the adaptability of proposed scheme is demonstrated by numerical experiments and shows that the present composite scheme offers better accuracy in comparison with other existing numerical methods.