隐式TVD格式在气液两相爆轰数值模拟中的应用

为研究FAE(燃料-空气炸药)爆炸参数规律,运用二维轴对称气液两相方程组模型,针对其中的气相方程组采用高分辨率的隐式TVD格式,液相方程组采用MacCormack格式,较好地对FAE气液两相爆轰产生的爆轰波的发展和传播过程进行了数值模拟,计算结果与国内外的研究结果符合良好.     

守恒型双曲方程的中心差分TVD格式研究

研究了如何利用迎风格式的耗散性构造中心差分ＴＶＤ格式的方法，给 相应的定理，构造出新的耗散表达式。新格式既保留了二阶中心差分格式灵活方便的优点，又吸收了迎风格式耗散项比较精细的特点，同时具有ＴＶＤ性质，使得新格式具有较同的激波分辨率。     

无波动,无自由参数,耗散的隐式差分格式

本文建立了求解NS方程和Euler方程无波动、无自由参数、耗散的隐式差分格式.该格式是TVD的和无条件稳定的.其隐式部分在1,2,3维情况下仅分别依赖于3,5,9个点,且系数矩阵是主对角占优的.计算例题表明,该方法可获得和显式方法相同的精度,能很好地捕捉激波和剪切层,且计算时间比显式有较多的节省.     

Central ADER schemes for hyperbolic conservation laws

In this paper we first briefly review the very high order ADER methods for solving hyperbolic conservation laws. ADER methods use high order polynomial reconstruction of the solution and upwind fluxes as the building block. They use a first order upwind Godunov and the upwind second order weighted average (WAF) fluxes. As well known the upwind methods are more accurate than central schemes. However, the superior accuracy of the ADER upwind schemes comes at a cost, one must solve exactly or approximately the Riemann problems (RP). Conventional Riemann solvers are usually complex and are not available for many hyperbolic problems of practical interest. In this paper we propose to use two central fluxes, instead of upwind fluxes, as the building block in ADER scheme. These are the monotone first order Lax-Friedrich (LXF) and the third order TVD flux. The resulting schemes are called central ADER schemes. Accuracy of the new schemes is established. Numerical implementations of the new schemes are carried out on the scalar conservation laws with a linear flux, nonlinear convex flux and non-convex flux. The results demonstrate that the proposed scheme, with LXF flux, is comparable to those using first and second order upwind fluxes while the scheme, with third order TVD flux, is superior to those using upwind fluxes. When compared with the state of art ADER schemes, our central ADER schemes are faster, more accurate, Riemann solver free, very simple to implement and need less computer memory. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws in one and two dimensions is presented.     

一类TVD型的迎风紧致差分格式

给出一种迎风型TVD(total variation diminishing)格式的构造方法,该方法通过限制器来抑制线性紧致格式在模拟间断流场时的非物理波动,可构造出非线性TVD型紧致格式(CTVD)．然后采用该法构造出了3阶和5阶的TVD型紧致格式,并通过模拟一维组合波和Riemann问题,二维激波-涡相互干扰和激波-边界层相互作用等来考察它们的性能．数值实验表明了该类格式的高阶精度和分辨率,且过间断基本无振荡．     

一类交错网格的Gauss型格式

本文在交错网格的情况下 ,利用 Gauss型求积公式构造了一类不需解 Riemann问题的求解一维单个双曲守恒律的二阶显式 Gauss型差分格式 ,证明了该格式在CFL条件限制下为 TVD格式 ,并证明了这类格式的收敛性 ,然后将格式推广到方程组的情形 .由于在交错网格的情况下构造的这类差分格式 ,不需要求解 Riemann问题 ,因此这类格式与诸如 Harten等的 TVD格式相比具有如下优点 :由于不需要完整的特征向量系 ,因此可用于求解弱双曲方程组 ,计算更快、编程更加简便等 .     

An efficient TVD‐WENO method for conservation laws

In this article, we present a high‐resolution hybrid scheme for solving hyperbolic conservation laws in one and two dimensions. In this scheme, we use a cheap fourth order total variation diminishing (TVD) scheme for smooth region and expensive seventh order weighted nonoscillatory (WENO) scheme near discontinuities. To distinguish between the smooth parts and discontinuities, we use an efficient adaptive multiresolution technique. For time integration, we use the third order TVD Runge‐Kutta scheme. The accuracy of the resulting hybrid high order scheme is comparable with these of WENO, but with significant decrease of the CPU cost. Numerical demonstrates that the proposed scheme is comparable to the high order WENO scheme and superior to the fourth order TVD scheme. Our scheme has the added advantage of simplicity and computational efficiency. Numerical tests are presented which show the robustness and effectiveness of the proposed scheme.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

高精度隐式参差光滑格式的构造

参照Lax-Wendroff格式的构造方法,就双曲型方程、抛物型方程和双曲-抛物型方程,构造了一种新的IRS(implicit residual smoothing)格式。该IRS格式有二阶或三阶时间精度且大大地拓宽了解的稳定区域和CFL数。这种新的IRS格式有中心加权型和迎风偏向型两种,并用von-Neumann分析方法分析了格式的稳定范围。讨论了在透平机械中广泛应用的Dawes方法的局限性,发现该方法对稳态问题得出的解与时间步长的选取有关,对粘性问题求解时,时间步长受严格限制。最后,结合TVD(total variation diminishing)格式和四阶Runge-Kutta技术,用IRS格式和Dawes方法对二维反射激波场进行了数值模拟,数值结果支持本文的分析结论。     

A TVD Type Wavelet-Galerkin Method for Hamilton-Jacobi Equations

In this paper,we use Daubechies scaling functions as test functions for the Galerkin method,and discuss Wavelet-Galerkin solutions for the Hamilton-Jacobi equations.It can be proved that the schemesare TVD schemes.Numerical tests indicate that the schemes are suitable for the Hamilton-Jacobi equations.Furthermore,they have high-order accuracy in smooth regions and good resolution of singularities.     

Second order scheme for scalar conservation laws with discontinuous flux

Burger, Karlsen, Torres and Towers in [9] proposed a flux TVD (FTVD) second order scheme with Engquist–Osher flux, by using a new nonlocal limiter algorithm for scalar conservation laws with discontinuous flux modeling clarifier thickener units. In this work we show that their idea can be used to construct FTVD second order scheme for general fluxes like Godunov, Engquist–Osher, Lax–Friedrich, … satisfying (A, B)-interface entropy condition for a scalar conservation law with discontinuous flux with proper modification at the interface. Also corresponding convergence analysis is shown. We show further from numerical experiments that solutions obtained from these schemes are comparable with the second order schemes obtained from the minimod limiter.     

Monotonicity criteria for difference schemes designed for hyperbolic equations

Previously formulated monotonicity criteria for explicit two-level difference schemes designed for hyperbolic equations (S.K. Godunov’s, A. Harten’s (TVD schemes), characteristic criteria) are extended to multileveled, including implicit, stencils. The characteristic monotonicity criterion is used to develop a universal algorithm for constructing high-order accurate nonlinear monotone schemes (for an arbitrary form of the desired solution) based on their analysis in the space of grid functions. Several new fourth-to-third-order accurate monotone difference schemes on a compact three-level stencil and nonexpanding (three-point) stencils are proposed for an extended system, which ensures their monotonicity for both the desired function and its derivatives. The difference schemes are tested using the characteristic monotonicity criterion and are extended to systems of hyperbolic equations.     

High-order accurate dissipative weighted compact nonlinear schemes

Based on the method deriving dissipative compact linear schemes (DCS), novel high-order dissipative weighted compact nonlinear schemes (DWCNS) are developed. By Fourier analysis,the dissipative and dispersive features of DWCNS are discussed. In view of the modified wave number, the DWCNS are equivalent to the fifth-order upwind biased explicit schemes in smooth regions and the interpolations at cell-edges dominate the accuracy of DWCNS. Boundary and near boundary schemes are developed and the asymptotic stabilities of DWCNS on both uniform and stretching grids are analyzed. The multi-dimensional implementations for Euler and Navier-Stokes equations are discussed. Several numerical inviscid and viscous results are given which show the good performances of the DWCNS for discontinuities capturing, high accuracy for boundary layer resolutions, good convergent rates (the root-mean-square of residuals approaching machine zero for solutions with strong shocks) and especially the damping effect on the spudous oscillations which were found in the solutions obtained by TVD and ENO schemes.     

NEW APPROACH TO THE LIMITER FUNCTIONS

1.IntroductionSince1980's,differenceschemeswithTVDorTVBpropertieshavebeenusedformoreandmoreCFDproblems,especiallythefollowingsystemofconservationlaws:ThereasonisthattheTVBpropertywillguaranteetheconvergenceofanysubsequenceofthedifferencesolutionsequencetoaweeksolutionofthedifferentialequation.Obviouslyiftheweeksolutionisunique,thenthewholesequencewillconvergetothatsolution.OneofthefrequentlyusedTVDschemeisthesecondorderfive--pointconservativeone:HereHi 1/2~H(Ull,,Uln,Uz71,U17,),iscons…     

The third-order relaxation schemes for hyperbolic conservation laws

A class of semi-discrete third-order relaxation schemes are presented for relaxation systems which approximate systems of hyperbolic conservation laws. These schemes for the scalar conservation law are shown to satisfy the property of total variation diminishing (TVD) in the zero relaxation limit. A third-order TVD Runge–Kutta splitting method is developed for the temporal discretization of the semi-discrete schemes. Numerical results are given illustrating these schemes on one-dimensional nonlinear problems.     

Total variation diminishing Runge-Kutta schemes

In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in a paper by Shu and Osher, suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.

     

非定常自由面流激波解的二阶守恒算法

将计算双曲型守恒律弱解的Lax-Wendroff型TVD格式推广到断面形状沿程任意变化的一般浅水方程组，构造了二阶精度的差分格式．新格式适用于模拟天然河道中溃坝洪水波的传播．提供了表明方法性能的算例，实际天然梯级水库溃坝问题的数值实验表明格式稳定，适应性强．     

The Entropy Condition for Implicit TVD Schemes

A class of implicit trapezoidal TVD schemes is proven to satisfy a discrete convex entropy inequality and the solution sequence of such implicit trapezoidal schemes converges to the physically relevant solution for genuinely nonlinear scalar conservation laws. The results are extended for a class of generalized implicit one-leg TVD schemes.     

TVD scheme for computing open channel wave flows

For the shallow water equations in the first approximation (Saint-Venant equations), a TVD scheme is developed for shock-capturing computations of open channel flows with discontinuous waves. The scheme is based on a special nondivergence approximation of the total momentum equation that does not involve integrals related to the cross-section pressure force and the channel wall reaction. In standard divergence difference schemes, most of the CPU time is spent on the computation of these integrals. Test computations demonstrate that the discontinuity relations reproduced by the scheme are accurate enough for actual discontinuous wave propagation to be numerically simulated. All the qualitatively distinct solutions for a dam collapsing in a trapezoidal channel with a contraction in the tailwater area are constructed as an example.     

求解Hamilton-Jacobi方程的有限元方法

本文将Galerkin二次有限元应于Hamilton-Jacobi方程，得到了求解Hamilton-Jacobi方程的数值格式。这些格式是TVD型的，在更强的条件下，基半离散格式的数值解收敛于Hamilton-Jacobi方程的粘性解。数值结果表明这类格式具有较高分辨导数间断的能力。     

An efficient WENO scheme for solving hyperbolic conservation laws

In this paper we propose a new WENO scheme, in which we use a central WENO [G. Capdeville, J. Comput. Phys. 227 (2008) 2977-3014] (CWENO) reconstruction combined with the smoothness indicators introduced in [R. Borges, M. Carmona, B. Costa, W. Sun Don, J. Comput. Phys. 227 (2008) 3191-3211] (IWENO). We use the central-upwind flux [A. Kurganov, S. Noelle, G. Petrova, SIAM J. Sci. Comp. 23 (2001) 707-740] which is simple, universal and efficient. For time integration we use the third order TVD Runge-Kutta scheme. The resulting scheme improves the convergence order at critical points of smooth parts of solution as well as decrease the dissipation near discontinuities. Numerical experiments of the new scheme for one and two-dimensional problems are reported. The results demonstrates that the proposed scheme is superior to the original CWENO and IWENO schemes.