Backward error analysis is an important tool to study long time behavior of numerical methods. The main idea of it is to use perturbed equations, namely modified equations, to represent the numerical solutions. Since stochastic backward error analysis has not been well developed so far. This paper investigates the stochastic modified equation and backward error analysis for Euler-Maruyama method with respect to strong convergence are built up. Like deterministic case, stochastic modified equations, expressed as formal series, do not converge in general. But there exists the optimal truncation of the series such that the one step error of the modified equations is sub-exponentially small with respect to time step. Moreover, the result of stochastic backward error analysis is used to study the error growth of the Euler-Maruyama method on Kubo oscillator.

In this paper, we consider the a posteriori error estimates of the finite volume element method for the general second-order quasilinear elliptic problems over a convex polygonal domain in the plane, propose a residual-based error estimator and derive the global upper and local lower bounds on the approximation error in the $H^1$-norm. Moreover, for some special quasilinear elliptic problems, we propose a residual-based a posteriori error estimator and derive the global upper bound on the error in the $L^2$-norm. Numerical experiments are also provided to verify our theoretical results.

We obtain improved derivative estimates for the solution of the semilinear singularly perturbed reaction-diffusion problem in one dimension. This enables us to modify the transition points between the fine and coarse parts of the Shishkin discretization mesh. We prove that the numerical solution, obtained by using the central finite-difference scheme on the modified mesh, retains the same order of convergence uniform in the perturbation parameter as on the standard Shishkin mesh. However, the modified mesh may be denser in the layers than the standard one, and, when this is the case, numerical results show an improvement in the accuracy of the computed solution.

The Crank-Nicolson (CN) orthogonal spline collocation method and its alternating direction implicit (ADI) counterpart are considered for the approximate solution of a class of linear parabolic problems in two space variables. It is proved that both methods are second order accurate in time and of optimal order in certain $H^j$ norms in space. Also, $L^∞$ estimates in space are derived.

In this paper, a local and parallel algorithm based on the multilevel discretization is proposed for solving the eigenvalue problem by the finite element method. With this new scheme, the eigenvalue problem solving in the finest grid is transferred to solutions of the eigenvalue problems on the coarsest mesh and a series of solutions of boundary value problems on each level mesh. Therefore this type of multilevel local and parallel method improves the overall efficiency of solving the eigenvalue problem. Some numerical experiments are presented to validate the efficiency of the new method.

We consider a fully discrete, efficient algorithm for magnetohydrodynamic (MHD) flow that is based on the Elsässer variable formulation and a timestepping scheme that decouples the MHD system but still provides unconditional stability with respect to the timestep. We prove stability and optimal convergence of the scheme, and also connect the scheme to one based on handling each decoupled system with a penalty-projection method. Numerical experiments are given which verify all predicted convergence rates of our analysis on some analytical test problems, show the results of the scheme on a set of channel flow problems match well the results found when the computation is done with MHD in primitive variable, and finally show the scheme performs well on a channel flow over a step.

Least squares solutions of linear equations $Ax=b$ are very important for parameter estimation in engineering, applied mathematics, and statistics. There are several methods for their solution including QR decomposition, Cholesky decomposition, singular value decomposition (SVD), and Krylov subspace methods. The latter methods were developed for sparse $A$ matrices that appear in the solution of partial differential equations. The QR (and its variant the RRQR) and the SVD methods are commonly used for dense $A$ matrices that appear in engineering and statistics. Although the Cholesky decomposition is backward stable and known to have the least operational count, several authors recommend the use of QR in applications. In this article, we take a fresh look at least squares problems for dense $A$ matrices with full column rank using numerical experiments guided by recent results from the theory of random matrices. Contrary to currently accepted belief, comparisons of the sensitivity of the Cholesky and QR solutions to random parameter perturbations for various low to moderate condition numbers show no significant difference to within machine precision. Experiments for matrices with artificially high condition numbers reveal that the relative difference in the two solutions is on average only of the order of $10^{-6}$. Finally, Cholesky is found to be markedly computationally faster than QR — the mean computational time for QR is between two and four times greater than Cholesky, and the standard deviation in computation times using Cholesky is about a third of that of QR. Our conclusion in this article is that for systems with $Ax=b$ where $A$ has full column rank, if the condition numbers are low or moderate, then the normal equation method with Cholesky decomposition is preferable to QR.

Numerical solutions of optimal Dirichlet boundary control problems for linear and semilinear wave equations are studied. The optimal control problem is reformulated as a system of equations (an optimality system) that consists of an initial value problem for the underlying (linear or semilinear) wave equation and a terminal value problem for the adjoint wave equation. The discretized optimality system is solved by a shooting method. The convergence properties of the numerical shooting method in the context of exact controllability are illustrated through computational experiments. In particular, in the case of the linear wave equation, convergent approximations are obtained for both smooth minimum $L^2$-norm Dirichlet control and generic, non-smooth minimum $L^2$-norm Dirichlet controls.

This paper presents the theory of $\mathcal{C}^1$-rational quadratic fractal interpolation surfaces (FISs) over a rectangular grid. First we approximate the original function along the grid lines of interpolation domain by using the univariate $\mathcal{C}^1$-rational quadratic fractal interpolation functions (fractal boundary curves). Then we construct the rational quadratic FIS as a blending combination with the $x$-direction and $y$-direction fractal boundary curves. The developed rational quadratic FISs are monotonic whenever the corresponding fractal boundary curves are monotonic. We derive the optimal range for the scaling parameters in both positive and negative directions such that the rational quadratic fractal boundary curves are monotonic in nature. The relation between $x$-direction and $y$-direction scaling matrices is deduced for symmetric rational quadratic FISs for symmetric surface data. The presence of scaling parameters in the fractal boundary curves helps us to get a wide variety of monotonic/symmetric rational quadratic FISs without altering the given surface data. Numerical examples are provided to demonstrate the comprehensive performance of the rational quadratic FIS in fitting a monotonic/symmetric surface data. The convergence analysis of the monotonic rational quadratic FIS to the original function is reported.

In this paper, an augmented immersed interface method has been developed for Helmholtz/Poisson equations on irregular domains in complex space. One of motivations of this paper is for simulations of wave scattering in different geometries. This paper is the first immersed interface method in complex space. The new method utilizes a combination of methodologies including the immersed interface method, a fast Fourier transform, augmented strategies, least squares interpolations, and the generalized minimal residual method (GMRES) for a Schur complement system, all in complex space. The new method is second order accurate in the $L^∞$ norm and requires $O(N log(N))$ operations. Numerical examples are provided for a variety of real or complex wave numbers.