An improvement of level set equations via approximation of a distance function

In the classical level set method, the slope of solutions can be very small or large, and it can make it difficult to get the precise level set numerically. In this paper, we introduce an improved level set equation whose solutions are close to the signed distance function to evolving interfaces. The improved equation is derived via approximation of the evolution equation for the distance function. Applying the comparison principle, we give an upper- and lower bound near the zero level set for the viscosity solution to the initial value problem.     

On convergence of finite volume schemes for one-dimensional two-phase flow in porous media

This paper deals with development and analysis of finite volume schemes for a one-dimensional nonlinear, degenerate, convection-diffusion equation having application in petroleum reservoir and groundwater aquifer simulation. The main difficulty is that the solution typically lacks regularity due to the degenerate nonlinear diffusion term. We analyze and compare three families of numerical schemes corresponding to explicit, semi-implicit, and implicit discretization of the diffusion term and a Godunov scheme for the advection term. L^∞ stability under appropriate CFL conditions and BV estimates are obtained. It is shown that the schemes satisfy a discrete maximum principle. Then we prove convergence of the approximate solution to the weak solution of the problem. Results of numerical experiments using the present approach are reported.     

一种无重新初始化的水平集方法的错觉轮廓捕捉新方法

在错觉轮廓捕捉模型建立前,我们要得到根据物体边界的符号距离函数时,用Eikonal方程不能实现的,我们用基于水平集方法的分割技术实现,扩大了模型的使用范围;在Zhu和Chan等人的错觉轮廓捕捉模型基础上引入了李纯明等人提出的符号距离约束信息,这就使得在水平集函数演化时不必对其重新初始化,并大大简化了模型的数值处理水平集函数的演化速度.并通过实验验证了该方法的优势.     

Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation

The paper deals with convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation. It is proved that the semi-implicit Euler method is convergent with strong order . The conditions under which the method is MS-stable and GMS-stable are determined and the numerical experiments are given.     

Level set based multi-scale methods for large deformation contact problems

We consider the numerical simulation of contact problems in elasticity with large deformations. The non-penetration condition is described by means of a signed distance function to the obstacle's boundary. Techniques from level set methods allow for an appropriate numerical approximation of the signed distance function preserving its non-smooth character. The emerging non-convex optimization problem subject to non-smooth inequality constraints is solved by a non-smooth multiscale SQP method in combination with a non-smooth multigrid method as interior solver. Several examples in three space dimensions including applications in biomechanics illustrate the capability of our methods.     

Two-derivative Runge–Kutta methods for PDEs using a novel discretization approach

We develop a novel and general approach to the discretization of partial differential equations. This approach overcomes the rigid restriction of the traditional method of lines (MOL) and provides flexibility in the treatment of spatial discretization. This method is essential for developing efficient numerical schemes for PDEs based on two-derivative Runge–Kutta (TDRK) methods, where the first and second derivatives must be discretized in an efficient way. This is unlikely to be achieved by using MOL. We then apply the explicit TDRK methods to the advection equations and analyze the numerical stability in the linear advection equation case. We conduct numerical experiments on the Burgers’ equation using the TDRK methods developed. We also apply a two-stage semi-implicit TDRK method of order-4 and stage-order-4 to the heat equation. A very significant improvement in the efficiency of this TDRK method is observed when compared to the popular Crank-Nicolson method. This paper is partially based on the work in Tsai’s PhD thesis (2011) [10].     

一种求解运动曲面上对流扩散方程的三维水平集方法

给出了一种求解运动曲面上对流扩散方程的三维水平集算法. 水平集函数被用来表示曲面.曲面上的微分方程及其解通过水平集方法被延拓到包含曲面的一个小邻域中. 一种半隐式的Crank-Nicholson 格式被用来做时间推进, 中心差分和三阶加权实质无振荡(WENO) 格式被分别用来离散方程中的扩散项和对流项. 分析证明了它在标准的Courant-Friedrichs-Lewy (CFL) 条件下的稳定性. 数值算例显示了它能取得二阶精度.     

Stability and consistency of the semi-implicit co-volume scheme for regularized mean curvature flow equation in level set formulation

We show stability and consistency of the linear semi-implicit complementary volume numerical scheme for solving the regularized, in the sense of Evans and Spruck, mean curvature flow equation in the level set formulation. The numerical method is based on the finite volume methodology using the so-called complementary volumes to a finite element triangulation. The scheme gives the solution in an efficient and unconditionally stable way.     

A numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection-diffusion equation

In this paper, a numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection diffusion equation is presented. The convergence and stability of the numerical approximation method are discussed by a new technique of Fourier analysis. The solvability of the numerical approximation method also is analyzed. Finally, applying Richardson extrapolation technique, a high-accuracy algorithm is structured and the numerical example demonstrated the theoretical results.     

The th moment stability for the stochastic pantograph equation

In this paper, we investigate the αth moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the Razumikhin technique. Especially the linear stochastic pantograph equations and the semi-implicit Euler method applying them are considered. The convergence result of the semi-implicit Euler method is obtained. The stability conditions of the analytic solution of those equations and the numerical method are given. Finally, some experiments are given.     

Numerical stability of higher-order derivative methods for the pantograph equation

Dealing with numerical stability of higher-order derivative methods with variable stepsize is the purpose of this paper for pantograph equations. A new way to compute this kind of equation is provided, and a sufficient condition for the numerical stability of high order derivative forms is given. Some numerical examples are presented to confirm our theoretical analysis.     

一维气液两相漂移模型全隐式AUSMV算法研究

气液两相漂移模型显式AUSMV（advection upstream splitting method combined with flux vector splitting method）算法的时间步长受限于CFL（Courant-Friedrichs-Lewy）条件，为了提高计算效率，建立了一种全隐式AUSMV算法求解气液两相漂移模型.采用AUSM格式结合FVS（flux vector splitting）格式构造连续方程和运动方程的对流项数值通量，AUSM格式构造压力项数值通量.离散控制方程是非线性方程组，采用六阶Newton(牛顿)法结合数值Jacobi矩阵求解.计算经典算例Zuber-Findlay激波管问题和复杂漂移关系变质量流动问题，结果分析表明:全隐式AUSMV算法，色散效应小，无数值震荡，计算精度高.在压力波波速高的条件下，可以显著提高计算效率，耗散效应小.     

一类强色散DGH方程的多辛普雷斯曼格式

DGH方程作为一类重要的非线性水波方程有着许多广泛的应用前景.基于Hamilton系统的多辛理论研究了一类强色散DGH方程的数值解法,利用多辛普雷斯曼方法构造了一种典型的半隐式的多辛格式.分析了该格式的局部能量和动量守恒律误差,并给出了数值算例.数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性.     

Multirate infinitesimal step methods for atmospheric flow simulation

The numerical solution of the Euler equations requires the treatment of processes in different temporal scales. Sound waves propagate fast compared to advective processes. Based on a spatial discretisation on staggered grids, a multirate time integration procedure is presented here generalising split-explicit Runge-Kutta methods. The advective terms are integrated by a Runge-Kutta method with a macro stepsize restricted by the CFL number. Sound wave terms are treated by small time steps respecting the CFL restriction dictated by the speed of sound.Split-explicit Runge-Kutta methods are generalised by the inclusion of fixed tendencies of previous stages. The stability barrier for the acoustics equation is relaxed by a factor of two.Asymptotic order conditions for the low Mach case are given. The relation to commutator-free exponential integrators is discussed. Stability is analysed for the linear acoustic equation. Numerical tests are executed for the linear acoustics and the nonlinear Euler equations.     

水平集方法与距离函数

讨论了有关水平集方法的基本问题,如保持为距离函数的方法,水平集方程解的存在性和唯一性。主要贡献是证明了,在距离函数约束下,水平集方程在初始零水平集附近有唯一解,它是关于演化界面的有向距离函数。并且用到了一些处理技巧:如注意到原始方程的任意解都是距离函数,将原始方程变化为另一简单形式。由于新的方程组不是一个经典方程组,则它被变换为一个普通形式,其中隐函数方法被采用。     

Newton iterations in implicit time-stepping scheme for differential linear complementarity systems

We propose a generalized Newton method for solving the system of nonlinear equations with linear complementarity constraints in the implicit or semi-implicit time-stepping scheme for differential linear complementarity systems (DLCS). We choose a specific solution from the solution set of the linear complementarity constraints to define a locally Lipschitz continuous right-hand-side function in the differential equation. Moreover, we present a simple formula to compute an element in the Clarke generalized Jacobian of the solution function. We show that the implicit or semi-implicit time-stepping scheme using the generalized Newton method can be applied to a class of DLCS including the nondegenerate matrix DLCS and hidden Z-matrix DLCS, and has a superlinear convergence rate. To illustrate our approach, we show that choosing the least-element solution from the solution set of the Z-matrix linear complementarity constraints can define a Lipschitz continuous right-hand-side function with a computable Lipschitz constant. The Lipschitz constant helps us to choose the step size of the time-stepping scheme and guarantee the convergence.     

Fully implicit time discretization for a free surface flow problem

One of the challenges in the numerics of free surface flows is the coupling of the flow field to the geometry of the domain. The most simple approach is an explicit decoupling, i.e. computing the flow field with geometrical information of a prior time step and then updating the geometry. This widely used approach leads to a severe CFL condition of the type , which may prescribe infinitesimally small time step sizes in the interesting case of a small Weber number (i.e. high surface tension). A semi-implicit approach utilizing the fact that , where x^k is a parametrization of the capillary boundary Γ, is also available [1]. This approach can be proven to be unconditionally stable but is of first order only. It also suffers from relatively strong numerical dissipation. We present a fully implicit approach using a backward differentiation formula to achieve a time discretization method that is of second order and only minimally dissipative. A numerical example of an oscillating drop showing very low numerical dissipation and second order convergence as well as numerical evidence for the stability of the method is presented. Since the method requires the solution of a highly nonlinear coupled system, possible preconditioners for this system are discussed, including a lower order decoupling. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

随机延迟微分方程的全隐式Euler方法

研究随机延迟微分方程数值解具有重要的意义,目前已有显式和半隐式两种数值方法,还没有全隐式的数值方法.本文构造了一种全隐式Euler方法,在该方法中用一些截断的随机变量代替维纳过程增量△W<,n>,接着证明了全隐式方法是1/2阶收敛的并通过数值实验验证了该方法的收敛性.最后,用数值实验表明在某些情况下全隐式方法的稳定性比半隐式方法好一些.     

A new stabilizing technique for boundary integral methods for water waves

Boundary integral methods to compute interfacial flows are very sensitive to numerical instabilities. A previous stability analysis by Beale, Hou and Lowengrub reveals that a very delicate balance among terms with singular integrals and derivatives must be preserved at the discrete level in order to maintain numerical stability. Such balance can be preserved by applying suitable numerical filtering at certain places of the discretization. While this filtering technique is effective for two-dimensional (2-D) periodic fluid interfaces, it does not apply to nonperiodic fluid interfaces. Moreover, using the filtering technique alone does not seem to be sufficient to stabilize 3-D fluid interfaces.

Here we introduce a new stabilizing technique for boundary integral methods for water waves which applies to nonperiodic and 3-D interfaces. A stabilizing term is added to the boundary integral method which exactly cancels the destabilizing term produced by the point vortex method approximation to the leading order. This modified boundary integral method still has the same order of accuracy as the point vortex method. A detailed stability analysis is presented for the point vortex method for 2-D water waves. The effect of various stabilizing terms is illustrated through careful numerical experiments.

     

T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise

The paper deals with the T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise. A difference equation is obtained by applying the numerical method to a linear test equation, in which the Wiener increment is approximated by a discrete random variable with two-point distribution. The conditions under which the method is T-stable are considered and the numerical experiments are given.