Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity

A time discretization method is called strongly stable(or monotone),if the norm of its numerical solution is nonincreasing.Although this property is desirable in various of contexts,many explicit Runge-Kutta(RK)methods may fail to preserve it.In this paper,we enforce strong stability by modifying the method with superviscosity,which is a numerical technique commonly used in spectral methods.Our main focus is on strong stability under the inner-product norm for linear problems with possibly non-normal operators.We propose two approaches for stabilization:the modified method and the filtering method.The modified method is achieved by modifying the semi-negative operator with a high order superviscosity term;the filtering method is to post-process the solution by solving a diffusive or dispersive problem with small superviscosity.For linear problems,most explicit RK methods can be stabilized with either approach without accuracy degeneration.Furthermore,we prove a sharp bound(up to an equal sign)on diffusive superviscosity for ensuring strong stability.For nonlinear problems,a filtering method is investigated.Numerical examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance.

Almost Sure Convergence Theorem and Strong Stability for Weighted Sums of NSD Random Variables

In this paper, Kolmogorov-type inequality for negatively superadditive dependent （NSD） random variables is established. By using this inequality, we obtain the almost sure convergence for NSD sequences, which extends the corresponding results for independent sequences and negatively associated （NA） sequences. In addition, the strong stability for weighted sums of NSD random variables is studied.

Strong Stability of Linear Forms in φ-Mixing Random Variables

In this paper some new results of strong stability of linear forms in φ-mixing random variables are given. It is mainly proved that for a sequence of φ-mixing random variables {xn,n≥1} and two sequences of positive numbers {an,n≥1} and {bn,n≥1} there exist d dn∈R,n = 1,2,..., such that bn^-1∑i=1^naixi-dn→0 a.s.under some suitable conditions. The results extend and improve the corresponding theorems for independent identically distributed random variables.

STRONG STABILITY PRESERVING PROPERTY OF THE DEFERRED CORRECTION TIME DISCRETIZATION

In this paper, we study the strong stability preserving （SSP） property of a class of deferred correction time discretization methods, for solving the method-of-lines schemes approximating hyperbolic partial differential equations.

HERMITE WENO SCHEMES WITH STRONG STABILITY PRESERVING MULTI-STEP TEMPORAL DISCRETIZATION METHODS FOR CONSERVATION LAWS

Based on the work of Shu [SIAM J. Sci. Stat. Comput, 9 （1988）, pp.1073-1084], we construct a class of high order multi-step temporal discretization procedure for finite volume Hermite weighted essential non-oscillatory （HWENO） methods to solve hyperbolic conservation laws. The key feature of the multi-step temporal discretization procedure is to use variable time step with strong stability preserving （SSP）. The multi-step tem- poral discretization methods can make full use of computed information with HWENO spatial discretization by holding the former computational values. Extensive numerical experiments are presented to demonstrate that the finite volume HWENO schemes with multi-step diseretization can achieve high order accuracy and maintain non-oscillatory properties near discontinuous region of the solution.

SOME LIMIT THEOREMS FOR SEQUENCES OF PAIRWISE NQD RANDOM VARIABLES

In this article, the authors study some limit properties for sequences of pairwise NQD random variables, which are not necessarily identically distributed. They obtain Baum and Katz complete convergence and the strong stability of Jamison＇s weighted sums for pairwise NQD random variables, which may have different distributions. Some wellknown results are improved and extended.

Direct discontinuous Galerkin method for the generalized Burgers-Fisher equation

In this study, we use the direct discontinuous Galerkin method to solve the generalized Burgers-Fisher equation. The method is based on the direct weak formulation of the Burgers-Fisher equation. The two adjacent cells are jointed by a numerical flux that includes the convection numerical flux and the diffusion numerical flux. We solve the ordinary differential equations arising in the direct Galerkin method by using the strong stability preserving Runge^Kutta method. Numerical results are compared with the exact solution and the other results to show the accuracy and reliability of the method.

A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Diffusion Problems

There have been many theoretical studies and numerical investigations of nonlocal diffusion(ND)problems in recent years.In this paper,we propose and analyze a new discontinuous Galerkin method for solving one-dimensional steady-state and time-dependent ND problems,based on a formulation that directly penalizes the jumps across the element interfaces in the nonlocal sense.We show that the proposed discontinuous Galerkin scheme is stable and convergent.Moreover,the local limit of such DG scheme recovers classical DG scheme for the corresponding local diff usion problem,which is a distinct feature of the new formulation and assures the asymptotic compatibility of the discretization.Numerical tests are also presented to demonstrate the eff ectiveness and the robustness of the proposed method.

High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form,typically used in implicit-explicit(IMEX)methods,is not possible.As shown in Boscarino et al.(J.Sci.Comput.68:975-1001,2016)for Runge-Kutta methods,these semi-implicit techniques give a great flexibility,and allow,in many cases,the construction of simple linearly implicit schemes with no need of iterative solvers.In this work,we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype lineal'advection-diffusion equation and in the setting of strong stability preserving(SSP)methods.Our findings are demonstrated on several examples,including nonlinear reaction-diffusion and convection-diffusion problems.

On the order of stable compensators for a class of time-delay system

The stabilization using a stable compensator does not introduce additional unstable zeros into the closed-loop transfer function beyond those of the original plant, so it is a desirable compensator, the price is that the compensator’s order will go up. This note considered the order of stable compensators for a class of time-delay systems. First, it is shown that for single-loop plants with at most one real right-half plane zero, a special upper bound for the minimal order of a strongly stabilizing compensator can be obtained in terms of the plant order; Second, it is shown that approximate unstable pole-zero cancellation does not occur, and the distances between distinct unstable zeroes are bounded below by a positive constant, then it is possible to find an upper bound for the minimal order of a strongly stabilizing compensator.

A Second-Order Implicit-Explicit Scheme for the Baroclinic-Barotropic Split System of Primitive Equations

The baroclinic-barotropic mode splitting technique is commonly employed in numerical solutions of the primitive equations for ocean modeling to deal with the multiple time scales of ocean dynamics.In this paper,a second-order implicit-explicit(IMEX)scheme is proposed to advance the baroclinic-barotropic split system.Specifically,the baroclinic mode and the layer thickness of fluid are evolved explicitly via the second-order strong stability preserving Runge-Kutta scheme,while the barotropic mode is advanced implicitly using the linearized Crank-Nicolson scheme.At each time step,the baroclinic velocity is first computed using an intermediate barotropic velocity.The barotropic velocity is then corrected by re-advancing the barotropic mode with an improved barotropic forcing.Finally,the layer thickness is updated by coupling the baroclinic and barotropic velocities together.In addition,numerical inconsistencies on the discretized sea surface height caused by the mode splitting are alleviated via a reconciliation process with carefully calculated flux deficits.Temporal truncation error is also analyzed to validate the second-order accuracy of the scheme.Finally,two benchmark tests from the MPAS-Ocean platform are conducted to numerically demonstrate the performance of the proposed IMEX scheme.

ODE-Based Multistep Schemes for Backward Stochastic Differential Equations

In this paper,we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations(BSDEs).By the nonlinear Feynman-Kac formula,we reformulate the BSDE into a pair of reference ordinary differential equations(ODEs),which can be directly discretized by many standard ODE solvers,yielding the corresponding numerical schemes for BSDEs.In particular,by applying strong stability preserving(SSP)time discretizations to the reference ODEs,we can propose new SSP multistep schemes for BSDEs.Theoretical analyses are rigorously performed to prove the consistency,stability and convergency of the proposed SSP multistep schemes.Numerical experiments are further carried out to verify our theoretical results and the capacity of the proposed SSP multistep schemes for solving complex associated problems.

Stability of Multiplier Ideal Sheaves

In the present article, the authors find and establish stability of multiplier ideal sheaves, which is more general than strong openness.

On Complete Hypersurfaces with Constant Mean Curvature and Finite L^p-norm Curvature in R^(n+1)

By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1＋1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n＋1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1＋1 are considered.

A Continuous State Feedback Controller Design for High-order Nonlinear Systems with Polynomial Growth Nonlinearities

In this paper, we investigate the problem of global stabilization for a general class of high-order and non-smoothly stabilizable nonlinear systems with both lower-order and higher-order growth conditions. The designed continuous state feedback controller is recursively constructed to guarantee the global strong stabilization of the closed-loop system.