A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem

This work is devoted to a posteriori error analysis of fully discrete finite element approximations to the time dependent Stokes system. The space discretization is based on popular stable spaces, including Crouzeix–Raviart and Taylor–Hood finite element methods. Implicit Euler is applied for the time discretization. The finite element spaces are allowed to change with time steps and the projection steps include alternatives that is hoped to cope with possible numerical artifices and the loss of the discrete incompressibility of the schemes. The final estimates are of optimal order in \(L^\infty (L^2) \) for the velocity error.     

On the discrete Riccati equation

Several bounds have been reported recently for the trace of the solution to the discrete algebraic matrix Riccati equation. This note adds an alternative one to them.     

On symplectic and multisymplectic schemes for the KdV equation

We examine some symplectic and multisymplectic methods for the notorious Korteweg-de Vries equation, with the question whether the added structure preservation that these methods offer is key in providing high quality schemes for the long time integration of nonlinear, conservative partial differential equations. Concentrating on second order discretizations, several interesting schemes are constructed and studied. Our essential conclusions are that it is possible to design very stable, conservative difference schemes for the nonlinear, conservative KdV equation. Among the best of such schemes are methods which are symplectic or multisymplectic. Semi-explicit, symplectic schemes can be very effective in many situations. Compact box schemes are effective in ensuring that no artificial wiggles appear in the approximate solution. A family of box schemes is constructed, of which the multisymplectic box scheme is a prominent member, which are particularly stable on coarse space-time grids.     

A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher's equation

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional time-fractional Fisher's equation, which is obtained from the standard one-dimensional Fisher's equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0<α<1). The proposed LDG is based on the LDG finite element method for space and finite difference method for time. We prove that the method is stable, and the numerical solution converges to the exact one with order O(h^k+1+τ^2?α), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The numerical experiments reveal that the LDG is very effective.     

On the discrete Lyapunov matrix equation

Some bounds for the arithmetic and the geometric means of the characteristic roots of the positive semidefinite solution to the discrete Lyapunov matrix equation are derived.     

On the discrete time algebraic Riccati equation

A detailed account of the properties of a class of algebraic Riccati equations which arise in discrete time control and filtering problems is given. It is shown that a generalized notion of detectability plays an important role in classifying solutions of these equations. This concept is also related to a minimum phase condition.     

On two linearized difference schemes for Burgers’ equation

Burgers’ equation can model several physical phenomena. In the first part of this work, we derive a three-level linearized difference scheme for Burgers’ equation, which is then proved to be energy conservative, unique solvable and unconditionally convergent in the maximum norm by the energy method combining with the inductive method. In the second part of the work, we prove the L_∞ unconditional convergence of a two-level linearized difference scheme for Burgers’ equation proposed by Sheng [A new difference scheme for Burgers equation, J. Jiangsu Normal Univ. 30 (2012), pp. 39–43], which was proved previously conditionally convergent.     

Difference schemes for the dispersive equation

In this paper a table of difference schemes for the dispersive equationu _i=au _xxx is presented. A collection of criterions for deriving stability conditions of difference schemes is given and applied to these difference schemes.     

On error estimates of the fully discrete penalty method for the viscoelastic flow problem

In this paper, a fully discrete finite element penalty method is presented for the two-dimensional viscoelastic flow problem arising in the Oldroyd model, in which the spatial discretization is based on the finite element approximation and the time discretization is based on the backward Euler scheme. Moreover, we provide the optimal error estimate for the numerical solution under some realistic assumptions. Finally, some numerical experiments are shown to illustrate the efficiency of the penalty method.     

On the existence of polynomial-time approximation schemes for the reoptimization of discrete optimization problems

It is shown that no polynomial-time approximation scheme exists for the reoptimization of the set covering problem in inserting an element into or eliminating it from any set. A similar result is obtained for the minimum graph coloring problem in inserting a vertex with at most two incidence edges and for the minimal bin packing problem in eliminating any element.     

On the eigenvalue-eigenvector method for solution of the stationary discrete matrix Riccati equation

The purpose of this correspondence is to point out that certain numerical problems encountered in the solution of the stationary discrete matrix Riccati equation by the eigenvalue-eigenvector method of Vanghan [1] can be avoided by a simple reformulation.     

Fitted strong stability-preserving schemes for the Black-Scholes-Barenblatt equation

We solve numerically a fully nonlinear Black–Scholes problem of Bellman type. The algorithm is focused on the so-called Delta greek, the first spatial derivative of the option price. Since the elliptic operator degenerates on the boundary we use a fitted finite volume discretization in space. Strong stability-preserving time-marching is further applied in accordance to the nonlinear nature of the differential problem. Numerical experiments validate our considerations.     

On the discrete time matrix Riccati equation of optimal control†

This paper is concerned with the discrete time matrix Riccati equation. The properties established are those of minimality, convergence, uniqueness and stability. Further the convergence of the policy space approximation technique is proved. These results are analogous to those known for the continuous-time Riccati equation, but the techniques used are simpler.     

Numerical schemes for the forward–backward heat equation

This paper is concerned with a class of forward–backward heat equations. We use Saulyev's scheme to formulate certain approximation schemes. Then a non-overlap domain decomposition method is presented for the numerical solution. The numerical experiments show that the given algorithm is feasible and effective.     

The extrapolated fully implicit scheme for the diffusion equation

The numerical solution of the diffusion equation with homogeneous boundary conditions is discussed with emphasis on the L-stable Lawson-Morris method (LMM) derived by the extrapolation of the fully implicit backward difference method. This type of scheme is particularly suitable when there is an initial/boundary discontinuity. In this paper, a scheme which improves the accuracy of LMM is proposed and discussed.     

A nonrecursive algebraic solution for the discrete Riccati equation

Equations for the optimal linear control and filter gains for linear discrete systems with quadratic performance criteria are widely documented. A nonrecursive algebraic solution for the Riccati equation is presented. These relations allow the determination of the steady-state solution of the Riccati equation directly without iteration. The relations also allow the direct determination of the transient solution for any particular time without proceeding recursively from the initial conditions. The method involves finding the eigenvalues and eigenvectors of the canonical state-costate equations.     

Lyapunov equation for infinite-dimensional discrete bilinear systems

Mean-square stability for discrete systems requires that uniform convergence is preserved between input and state correlation sequences. Such a convergence preserving property holds for an infinite-dimensional bilinear system if and only if the associate Lyapunov equation has a unique strictly positive solution.     

New schemes for the coupled nonlinear Schrödinger equation

In this paper, we present three new schemes for the coupled nonlinear Schrödinger equation. The three new schemes are multi-symplectic schemes that preserve the intrinsic geometry property of the equation. The three new schemes are also semi-explicit in the sense that they need not solve linear algebraic equations every time-step, which is usually the most expensive in numerical simulation of partial differential equations. Many numerical experiments on collisions of solitons are presented to show the efficiency of the new multi-symplectic schemes.