In this article we study an approximate model for a binary fluid flow in a three-dimensional bounded domain. The governing equations consist of the Allen–Cahn equation for the order (phase) parameter $\phi$ coupled with the Navier–Stokes-$α$ (NS-$α$) system for the velocity $u$. We discretize these equations in time using the implicit Euler scheme and we prove that the global attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.

We present an approximate algorithm to solve only one variable out of a linear system defined by a matrix with off-diagonal exponential decay entries (including the practically most important class of band limited matrices) via a sub-linear system. This approach thus enables us to solve any subset of solution variables. Parallel implementation of such approximate schemes for every variable enables us to solve the linear system with computational time independent of the matrix size.

Spread option contracts are becoming increasingly important, as they frequently arise in the energy derivative markets, e.g. exchange electricity for oil. In this paper, we study the pricing of European and American spread options. We consider the two-dimensional Black-Scholes Partial Differential Equation (PDE), use finite difference discretization in space and consider Crank-Nicolson (CN) and Modified Craig-Sneyd (MCS) Alternating Direction Implicit (ADI) methods for timestepping. In order to handle the early exercise feature arising in American options, we employ the discrete penalty iteration method, introduced and studied in Forsyth and Vetzal (2002), for one-dimensional PDEs discretized in time by the CN method. The main novelty of our work is the incorporation of the ADI method into the discrete penalty iteration method, in a highly efficient way, so that it can be used for two or higher-dimensional problems. The results from spread option pricing are compared with those obtained from the closed-form approximation formulae of Kirk (1995), Venkatramanan and Alexander (2011), Monte Carlo simulations, and the Brennan-Schwartz ADI Douglas-Rachford method, as implemented in MATLAB. In all spread option test cases we considered, including American ones, our ADI-MCS method, implemented on appropriate non-uniform grids, gives more accurate prices and Greeks than the MATLAB ADI method.

Data envelopment analysis (DEA), as originally proposed, is a methodology for evaluating the relative efficiencies of peer decision making units (DMUs) under some general assumptions. DEA models with non-homogeneous DMUs and multi-activity structures are two different subjects referring to relaxing various assumptions. In this paper, we show that these two formulations are both derived by embedding the corresponding process into a general parallel DEA model. Furthermore, following the parallel DEA framework, general DEA models for multi-activity and non-homogeneity are proposed, which are able to handle many situations where different aspects of non-homogeneity or multi-activities exist. This study provides important insights into the existing DEA models for non-homogeneity and multi-activity.

A piecewise smooth numerical approximation should be in some sense as smooth as its target function in order to have the optimal order of approximation measured in Sobolev norms. In the context of discontinuous finite element approximation, that means the shape function needs to be numerically smooth in the interiors as well as across the interfaces of elements. In previous papers [2, 8] we defined the concept of numerical smoothness and stated the principle: numerical smoothness is necessary for optimal order convergence. We proved this principle for discontinuous piecewise polynomials on $\mathbb{R}^n$, $1 ≤ n ≤ 3$. In this paper, we generalize it to include discontinuous piecewise non-polynomial functions, e.g., rational functions, on quadrilateral subdivisions whose pullbacks are polynomials such as bilinears, bicubics and so on.

A group of high order Gautschi-type exponential wave integrators (EWIs) Fourier pseudospectral method are proposed and analyzed for solving the nonlinear Klein-Gordon equation (KGE) in the nonrelativistic limit regime, where a parameter $0 < ε\ll 1$ which is inversely proportional to the speed of light, makes the solution propagate waves with wavelength $O(ε^2)$ in time and $O(1)$ in space. With the Fourier pseudospectral method to discretize the KGE in space, we propose a group of EWIs with designed Gautschi’s type quadratures for the temporal integrations, which can offer any intended even order of accuracy provided that the solution is smooth enough, while all the current existing EWIs offer at most second order accuracy. The scheme is explicit, time symmetric and rigorous error estimates show the meshing strategy of the proposed method is time step $\tau = O(ε^2)$ and mesh size $h = O(1)$ as $0 < ε \ll 1$, which is ‘optimal’ among all classical numerical methods towards solving the KGE directly in the limit regime, and which also distinguish our methods from other high order approaches such as Runge-Kutta methods which require $\tau = O(ε^3)$. Numerical experiments with comparisons are done to confirm the error bound and show the superiority of the proposed methods over existing classical numerical methods.

This paper studies a regularization approach for simultaneously reconstructing space-time dependent Robin coefficient $γ(\rm x, t)$ and heat flux $q(\rm x, t)$. The differentiability results and adjoint systems are established. A standard finite element method (FEM) is employed to discretize the constrained optimization problem which is reduced to a sequence of unconstrained optimization problem by adding regularization terms. We propose an improved algorithm for the introduced problem based on modified conjugate gradient method (MCGM) for quadratic minimization. Numerical experiments present the efficiency, accuracy, and robustness of the proposed algorithm.

A general framework for the regularization of constrained PDEs, also called operator differential-algebraic equations (operator DAEs), is presented. The given procedure works for semi-explicit and semi-linear operator DAEs of first order including the Navier-Stokes and other flow equations. The proposed reformulation is consistent, i.e., the solution of the PDE remains untouched. Its main advantage is that it regularizes the operator DAE in the sense that a semi-discretization in space leads to a DAE of lower index. Furthermore, a stability analysis is presented for the linear case, which shows that the regularization provides benefits also for the application of the Rothe method. For this, the influence of perturbations is analyzed for the different formulations. The results are verified by means of a numerical example with an adaptive space discretization.