Total Variation Diminishing Implicit Runge-Kutta Methods for Dissipative Advection-Diffusion Problems in Astrophysics

We investigate the properties of dissipative full discretizations for the equations of motion associated with models of flow and radiative transport inside stars. We derive dissipative space discretizations and demonstrate that together with specially adapted total-variation-diminishing (TVD) or strongly stable Runge-Kutta time discretizations with adaptive step-size control this yields reliable and efficient integrators for the underlying high-dimensional nonlinear evolution equations. For the most general problem class, fully implicit SDIRK methods are demonstrated to be competitive when compared to popular explicit Runge-Kutta schemes as the additional effort for the solution of the associated nonlinear equations is compensated by the larger step-sizes admissible for strong stability and dissipativity. For the parameter regime associated with semiconvection we can use partitioned IMEX Runge-Kutta schemes, where the solution of the implicit part can be reduced to the solution of an elliptic problem. This yields a significant gain in performance as compared to either fully implicit or explicit time integrators. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On first- and second-order difference schemes for differential-algebraic equations of index at most two

Difference schemes of the Euler and trapezoidal types for the numerical solution of the initial-value problem for linear differential-algebraic equations are examined. These schemes are analyzed for model examples, and their superiority over the familiar first- and second-order implicit methods is shown. Conditions for the convergence of the proposed algorithms are formulated.     

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.     

求解双曲型方程的一致二阶守恒型差分格式的构造及数值实验

由于TVD格式具有激波分辨率高与非物理振荡小的特点,在气体动力学问题的求解中得到了广泛的应用.但现有的TVD格式受其构造方式所限,在解的局部极值点附近只能达到一阶精度. 考虑以下单个双曲型方程:     

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.     

一类TVD格式的熵强迫函数及熵条件

1.引言 在拟线性双曲型方程差分方法的研究中,数值解的收敛性是一个重要的问题,因为这一性质决定了数值解能否近似地反映真实的物理现象。数值解的收敛性实际上包含了三方面的内容: 1°当网格步长趋于零时,数值解序列包含一个按某种范数收敛到函数u的子序列。     

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.     

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。     

Product integration methods for solving a system of nonlinear Volterra integral equations

In this paper the technique of subtracting out singularities is used to derive explicit and implicit product Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of Volterra integral equations with a weakly singular kernel. The convergence proofs of the numerical schemes are presented; these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global Lipschitz condition.     

On the convergence of high resolution methods with multiple time scales for hyperbolic conservation laws

A class of finite volume methods based on standard high resolution schemes, but which allows spatially varying time steps, is described and analyzed. A maximum principle and the TVD property are verified for general advective flux, extending the previous theoretical work on local time stepping methods. Moreover, an entropy condition is verified which, with sufficient limiting, guarantees convergence to the entropy solution for convex flux.

     

Second-order schemes for conservation laws with discontinuous flux modelling clarifier–thickener units

Continuously operated clarifier–thickener (CT) units can be modeled by a non-linear, scalar conservation law with a flux that involves two parameters that depend discontinuously on the space variable. This paper presents two numerical schemes for the solution of this equation that have formal second-order accuracy in both the time and space variable. One of the schemes is based on standard total variation diminishing (TVD) methods, and is addressed as a simple TVD (STVD) scheme, while the other scheme, the so-called flux-TVD (FTVD) scheme, is based on the property that due to the presence of the discontinuous parameters, the flux of the solution (rather than the solution itself) has the TVD property. The FTVD property is enforced by a new nonlocal limiter algorithm. We prove that the FTVD scheme converges to a BV _t solution of the conservation law with discontinuous flux. Numerical examples for both resulting schemes are presented. They produce comparable numerical errors, while the FTVD scheme is supported by convergence analysis. The accuracy of both schemes is superior to that of the monotone first-order scheme based on the adaptation of the Engquist–Osher scheme to the discontinuous flux setting of the CT model (Bürger, Karlsen and Towers in SIAM J Appl Math 65:882–940, 2005). In the CT application there is interest in modelling sediment compressibility by an additional strongly degenerate diffusion term. Second-order schemes for this extended equation are obtained by combining either the STVD or the FTVD scheme with a Crank–Nicolson discretization of the degenerate diffusion term in a Strang-type operator splitting procedure. Numerical examples illustrate the resulting 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.     

Modified Upwind Taylor Finite Element Schemes for 1-D Conservation Laws I. A Basic Idea

In this paper, first, modified upwind finite element schemes are presented for two-point value problem, and then a class of modified upwind Taylor finite element schemes are derived for one dimensional linear hyperbolic equation. The main point of the paper is how to consider the upwind property to construct base functions to make the schemes obtained be MmB (or TVD). Numerical experiments are given to show that the method is efficient to solve the discontinuous solutions.     

ON THE CONVERGENCE OF IMPLICIT DIFFERENCE SCHEMES FOR HYPERBOLIC CONSERVATION LAWS

1. IntroductionWe are are interested in the fOllowing Cauchy problem for scalar conservation lawswhere the initial data uo E BV(R) and the flux function f 6 C'(n).It is well known that this problem may not always have a smooth global solution even ifthe i…     

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．     

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。     

Centred TVD schemes for hyperbolic conservation laws

New first- and high-order centred methods for conservation lawsare presented. Convenient TVD conditions for constructing centredTVD schemes are then formulated and some useful results areproved. Two families of centred TVD schemes are constructedand extended to nonlinear systems. Some numerical results arealso presented.     

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 .     

Exact finite-difference methods based on nonlinear means

The necessary and sufficient condition for an autonomous, ordinary differential equation to be exactly solved by a given Evans-Sanugi, nonlinear, one-step, finite difference method based on ‘classical means’ are derived. A necessary, but not sufficient, condition yields the most general nonlinearity for the differential equation which is independent of the step size. Examples of differential equations for which either nonlinear trapezoidal or nonlinear implicit midpoint methods based on arithmetic, harmonic, contraharmonic, quadratic, geometric, Heronian, centroidal and logarithmic means are exact, are presented. These new exact difference schemes may be useful in future developments of new Denk-Bulirsch, Le-Roux or Kojouhavov-Chen schemes for nonlinear evolution equations with or without blow-up.     

Smoothing effects on the IMR and ITR

This paper is a study of the effects of smoothing on the implicit midpoint rule (IMR) and the implicit trapezoidal rule (ITR) with implications for extrapolation of the numerical solution of ordinary differential equations. We extend the study of the well-known smoothing formula of Gragg to a two-step smoothing formula and compare the effectiveness of their use with the IMR and ITR for nonstiff and strongly stiff cases. We present an analysis of the Prothero-Robinson problem and as well as experimental results on linear and nonlinear problems.