Quadric SFDI for Laplacian Discretisation in Lagrangian Meshless Methods

In the Lagrangian meshless(particle)methods,such as the smoothed particle hydrodynamics(SPH),moving particle semi-implicit(MPS)method and meshless local Petrov-Galerkin method based on Rankine source solution(MLPG_R),the Laplacian discretisation is often required in order to solve the governing equations and/or estimate physical quantities(such as the viscous stresses).In some meshless applications,the Laplacians are also needed as stabilisation operators to enhance the pressure calculation.The particles in the Lagrangian methods move following the material velocity,yielding a disordered(random)particle distribution even though they may be distributed uniformly in the initial state.Different schemes have been developed for a direct estimation of second derivatives using finite difference,kernel integrations and weighted/moving least square method.Some of the schemes suffer from a poor convergent rate.Some have a better convergent rate but require inversions of high order matrices,yielding high computational costs.This paper presents a quadric semi-analytical finite-difference interpolation(QSFDI)scheme,which can achieve the same degree of the convergent rate as the best schemes available to date but requires the inversion of significant lower-order matrices,i.e.3×3 for 3D cases,compared with 6×6 or 10×10 in the schemes with the best convergent rate.Systematic patch tests have been carried out for either estimating the Laplacian of given functions or solving Poisson’s equations.The convergence,accuracy and robustness of the present schemes are compared with the existing schemes.It will show that the present scheme requires considerably less computational time to achieve the same accuracy as the best schemes available in literatures,particularly for estimating the Laplacian of given functions.

Three Kinds of Discrete Formulae for the Caputo Fractional Derivative

In this paper,three kinds of discrete formulae for the Caputo fractional derivative are studied,including the modified L1 discretisation forα∈(O,1),and L2 discretisation and L2C discretisation forα∈(1,2).The truncation error estimates and the properties of the coeffcients of all these discretisations are analysed in more detail.Finally,the theoretical analyses areverifiedby thenumerical examples.

Stability analysis for time delay control of nonlinear systems in discrete-time domain with a standard discretisation method

This paper provides stability analysis results for discretised time delay control(TDC)as implemented in a sampled data system with the standard form of zero-order hold.We first substantiate stability issues in discrete-time TDC using an example and propose sufficient stability criteria in the sense of Lyapunov.Important parameters significantly affecting the overall system stability are the sampling period,the desired trajectory and the selection of the reference model dynamics.

INTEGRABLE DISCRETISATION OF THE LOTKA-VOLTERRA SYSTEM

In this paper, we apply Hirota＇s discretisation to a three-dimensional integrable Lotka- Volterra system. By analyzing the three-dimensional modified equation of the resulting numerical method, we show that it is volume-preserving, and has two independent first integrals. Moreover, it can be formally reduced to a system in one dimension via a volume- preserving transformation. If the given initial value is located in the positive octant, we prove that the numerical solution is confined to a one-dimensional connected and compact space which is diffeomorphic to a circle.

On the Fractional Derivatives with an Exponential Kernel

The present article mainly focuses on the fractional derivatives with an exponential kernel(“exponential fractional derivatives”for brevity).First,several extended integral transforms of the exponential fractional derivatives are proposed,including the Fourier transform and the Laplace transform.Then,the L2 discretisation for the exponential Caputo derivative with a∈(1,2)is established.The estimation of the truncation error and the properties of the coefficients are discussed.In addition,a numerical example is given to verify the correctness of the derived L2 discrete formula.

The State Equations Methods for Stochastic Control Problems

The state equations of stochastic control problems,which are controlled stochastic differential equations,are proposed to be discretized by the weak midpoint rule and predictor-corrector methods for the Markov chain approximation approach. Local consistency of the methods are proved.Numerical tests on a simplified Merton's portfolio model show better simulation to feedback control rules by these two methods, as compared with the weak Euler-Maruyama discretisation used by Krawczyk.This suggests a new approach of improving accuracy of approximating Markov chains for stochastic control problems.

Discretization behavior analysis of a switching control system from a unified mathematical approach

A useful unified analysis framework is proposed for exploring the intriguing behaviors of a second-order switching control system. Complex discretization behaviors of the switching control system are explored in detail, and some intrinsic relationships between the system periodic behaviors and their associated symbolic sequences are studied. Keywords Switching control - Chaos - Discretisation - Periodicity This work was supported by the Australian Research Council and the Hong Kong Research Grants Council for their financial supports, under the CERG Grants CityU 1018/01E, 1004/02E, and 1115/03E.

Approximating Stationary Statistical Properties

It is well-known that physical laws for large chaotic dynamical systems are revealed statistically.Many times these statistical properties of the system must be approximated numerically.The main contribution of this manuscript is to provide simple and natural criterions on numerical methods (temporal and spatial discretization) that are able to capture the stationary statistical properties of the underlying dissipative chaotic dynamical systems asymptotically.The result on temporal approximation is a recent finding of the author,and the result on spatial approximation is a new one.Applications to the infinite Prandtl number model for convection and the barotropic quasi-geostrophic model are also discussed.

A Geometric Space-Time Multigrid Algorithm for the Heat Equation

We study the time-dependent heat equation on its space-time domain that is discretised by a k-spacetree.k-spacetrees are a generalisation of the octree concept and are a discretisation paradigm yielding a multiscale representation of dynamically adaptive Cartesian grids with low memory footprint.The paper presents a full approximation storage geometric multigrid implementation for this setting that combines the smoothing properties of multigrid for the equation’s elliptic operator with a multiscale solution propagation in time.While the runtime and memory overhead for tackling the all-in-one space-time problem is bounded,the holistic approach promises to exhibit a better parallel scalability than classical time stepping,adaptive dynamic refinement in space and time fall naturally into place,as well as the treatment of periodic boundary conditions of steady cycle systems,on-time computational steering is eased as the algorithm delivers guesses for the solution’s long-term behaviour immediately,and,finally,backward problems arising from the adjoint equation benefit from the the solution being available for any point in space and time.

CONVERGENCE OF NUMERICAL SCHEMES FOR A CONSERVATION EQUATION WITH CONVECTION AND DEGENERATE DIFFUSION

The approximation of problems with linear convection and degenerate nonlinear difFusion,which arise in the framework of the transport of energy in porous media with thermodynamic transitions,is done usingθ-scheme based on the centred gradient discretisation method.The convergence of the numerical scheme is proved,although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field,owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting.Some numerical examples,using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme,show the role ofθfor stabilising the scheme.

A FRAMEWORK OF PARALLEL ALGEBRAIC MULTILEVEL PRECONDITIONING ITERATIONS

A framework for parallel algebraic multilevel preconditioning methods presented for solving large sparse systems of linear equstions with symmetric positive definite coefficient matrices,which arise in suitable finite element discretizations of many second-order self-adjoint elliptic boundary value problems. This framework not only covers all known parallel algebraic multilevel preconditioning methods, but also yields new ones. It is shown that all preconditioners within this framework have optimal orders of complexities for problems in two-dimensional(2-D) and three-dimensional (3-D) problem domains, and their relative condition numbers are bounded uniformly with respect to the numbers of both levels and nodes.

Observer-based fault detection for switched systems with all unstable subsystems

This paper deals with the fault detection observer design problem for switched system with all modes unstable.First,the error system model is presented.With the consideration of the instability caused by the unobservable condition of(Ai,Ci),a switching signal is obtained via average dwell time method and discretised Lyapunov function.By means of linear matrix inequalities,sufficient conditions pledging exponential stability and H∞performance are proposed.Through transferring the observer design problem into H∞performance analysis,proper observer gain is obtained via solving these LMIs.In the end,the validity of the proposed Luenberger observer is verified by an example.