Numerical performance of parallel group explicit solvers for the solution of fourth order elliptic equations

Many applications in applied mathematics and engineering involve numerical solutions of partial differential equations (PDEs). Various discretisation procedures such as the finite difference method result in a problem of solving large, sparse systems of linear equations. In this paper, a group iterative numerical scheme based on the rotated (skewed) five-point finite difference discretisation is proposed for the solution of a fourth order elliptic PDE which represents physical situations in fluid mechanics and elasticity. The rotated approximation formulas lead to schemes with lower computational complexities compared to the centred approximation formulas since the iterative procedure need only involve nodes on half of the total grid points in the solution domain. We describe the development of the parallel group iterative scheme on a cluster of distributed memory parallel computer using Message-Passing Interface (MPI) programming environment. A comparative study with another group iterative scheme derived from the centred difference formula is also presented. A detailed performance analysis of the parallel implementations of both group methods will be reported and discussed.     

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This paper deals with a variant Boussinesq equations which describes the propagation of shallow water waves in a lake or near an ocean beach. We derive out two hetero-B\"{a}cklund transformations between the variant Boussinesq equations and two linear parabolic equations by using the extended homogeneous balance method. We also obtain two hetero-B\"{a}cklund transformations between the variant Boussinesq equations and Burgers equations. Furthermore, we obtain two hetero-B\"{a}cklund transformation between the variant Boussinesq equations and heat equations. By using these B\"{a}cklund transformations and so-called "seed solution", we obtain a large number of explicit exact solutions of the variant Boussinesq equations. Especially, The infinite explicit exact singular wave solutions of variant Boussinesq equations are obtained for the first time. It is worth noting that these singular wave solutions of variant Boussinesq equations will blow up on some lines or curves in the $(x,t)$ plane. These facts reflect the complexity of the structure of the solution of variant Boussinesq equations. It also reflects the complexity of shallow water wave propagation from one aspect.     

Explicit peaked wave solutions to the generalized Camassa-Holm equation

By constructing auxiliary differential equations, we obtain peaked solitary wave solutions of the generalized Camassa-Holm equation, including periodic cusp waves expressed in terms of elliptic functions.     

High‐order semi‐implicit time‐integrators for a triangular discontinuous Galerkin oceanic shallow water model

We extend the explicit in time high‐order triangular discontinuous Galerkin (DG) method to semi‐implicit (SI) and then apply the algorithm to the two‐dimensional oceanic shallow water equations; we implement high‐order SI time‐integrators using the backward difference formulas from orders one to six. The reason for changing the time‐integration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for high‐order DG methods. Changing the time‐integration method to SI allows one to circumvent the stability criterion due to the gravity waves, which for most shallow water applications are the fastest waves in the system (the exception being supercritical flow where the Froude number is greater than one). The challenge of constructing a SI method for a DG model is that the DG machinery requires not only the standard finite element‐type area integrals, but also the finite volume‐type boundary integrals as well. These boundary integrals pose the biggest challenge in a SI discretization because they require the construction of a Riemann solver that is the true linear representation of the nonlinear Riemann problem; if this condition is not satisfied then the resulting numerical method will not be consistent with the continuous equations. In this paper we couple the SI time‐integrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: high‐order accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of time‐explicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new high‐order SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem. The results show that the new high‐order SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use high‐order (HO) time‐integrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with low‐order time discretizations. Published in 2009 by John Wiley & Sons, Ltd.     

Explicit finite volume non‐oscillatory schemes for 2D transient free‐surface flows

A class of high‐resolution non‐oscillatory shock‐capturing Roe, TVD and ENO explicit schemes in finite volume approach are presented for the computation of 2D unsteady rapidly varied open channel flows. In order to apply these schemes to simulate the hydraulic phenomena in field, the Strang‐type operator splitting technique is adopted to treat the flow with bottom slope and friction terms. Verifications of the proposed schemes are made by comparison with analytical solutions or experimental data, and very good agreements are obtained. To illustrate the efficiency and stability of the present algorithms, four typical problems of rapidly varied flows are solved and the results of different schemes are compared. It is demonstrated that the proposed method is accurate, robust and highly stable even in the flows with very strong discontinuites, which need no tuning of any adjustable parameter, such as artificial viscosity coefficient, as other methods do, and is a reliable mathematical modeling for 2D practical hydraulic engineering applications. Copyright © 1999 John Wiley & Sons, Ltd.     

ACOUSTIC–MEAN FLOW INTERACTION AND VORTEX SHEDDING IN SOLID ROCKET MOTORS

The present work is devoted to the numerical simulation of two important phenomena in the field of solid propellant rocket motors: the first is acoustic boundary layers that develop above the burning propellant; the other is a periodic vortex-shedding phenomenon which is the result of a strong coupling between the instability of mean flow shear layers and acoustic motions in the chamber. To predict the acoustic boundary layer, computations were performed for the lower half of a rectangular chamber with bottom-side injection. The outflow pressure is sinusoidally perturbed at a given frequency. For the highest CFL numbers the implicit scheme is not able to compute the unsteadiness in the acoustic boundary layer. With very low CFL numbers or with the explicit scheme the main features of the acoustic field are captured. To simulate the vortex-shedding mechanismin a segmented solid rocket motor, the explicit version is used. This computation shows a mechanism for ‘self-excited’ vortex shedding close to the second axial mode frequency. The use of the flux-splitting technique reduces substantially the amplitude of the oscillations. A few iterations are done with flux splitting, then the computation is performed without this technique. In this case both the frequency and the intensity are well predicted. A geometry more representative of the solid rocket motor is also computed. In this case the vortex-shedding process is more complex and pairing is observed.     

色散方程的一类具任意稳定性的显格式

本文对色散方程u_r=au_xxx构造了一类中间层包含四个结点,带两个参数m和θ的三层显式差分格式。当m和θ满足一定的关系时,其稳定性条件为|γ|≤(m+1)/(4(m-1))(|m|>1),从而当取m充分接近1时,可得到任意大的稳定性条件,并且保持截断误差阶不变。数值例子验证了理论分析的结果。     

Four-point explicit difference schemes for the dispersive equation

A class of three-level explicit difference schemes for the dispersive equationu_1=au_(xxx)are established These schemes have higher stability and involve four meshpoints at the middle level.Their local truncation errors are O(τ+h)and stabilityconditions are from|R|≤0.25 to|R|≤10,where|R|=|a|τ/h~3,which is muchbetter than|R|≤0.25.     

一类系数依赖于两个参数的齐次递推式之解

本文拓广了文献［１］的研究范围,给出了一类依赖于两个参数的变系数递推式的解的明显表达式．有关结果,使得象Ｌａh数,两类Ｓｔｉｒｌｉｎｇ数以及置换群中的置换个数等多方面的相应定解问题,皆可以直接解出．这对具大数值双指标的递推式的计算,亦或在其理论的研究方面,都有其作用．     

二维抛物型方程的一族高精度分支稳定显格式

1 引言 在渗流、扩散、热传导等领域中经常会遇到求解二维抛物型方程的初边值问题 {(6)u/(6)=a((6)2u/(6)x2+(6)2u/(6)y2), 0＜x,y＜L,t＞0,a＞0u(x, y, 0) =φ(x, y), 0 ≤ x, y ≤ L (1)u(0,y,t) =f1(y,t),u(L,y,t) =f2...