A new coupled high-order compact method for the three-dimensional nonlinear biharmonic equations

This paper presents a new family of fourth- compact finite difference schemes for the numerical solution of three-dimensional nonlinear biharmonic equations using coupled approach. The numerical solutions of unknown variable and its first- derivatives as well as v(=Δ u) are obtained not only in the interior but also at the boundary. A prominent contribution of this work is that the boundary conditions for the variable v are approximated more accurately, which plays an important role for the efficiency of calculation. Finally, numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these schemes, including the steady Navier–Stokes equation in terms of vorticity-stream function formulation.     

Pseudospectral methods based on nonclassical orthogonal polynomials for solving nonlinear variational problems

Two direct pseudospectral methods based on nonclassical orthogonal polynomials are proposed for solving finite-horizon and infinite-horizon variational problems. In the proposed finite-horizon and infinite-horizon methods, the rate variables are approximated by the Nth degree weighted interpolant, using nonclassical Gauss-Lobatto and Gauss points, respectively. Exponential Freud type weights are introduced for both of nonclassical orthogonal polynomials and weighted interpolation. It is shown that the absolute error in weighted interpolation is dependent on the selected weight, and the weight function can be tuned to improve the quality of the approximation. In the finite-horizon scheme, the functional is approximated based on Gauss-Lobatto quadrature rule, thereby reducing the problem to a nonlinear programming one. For infinite-horizon problems, an strictly monotonic transformation is used to map the infinite domain onto a finite interval. We transcribe the transformed problem to a nonlinear programming using Gauss quadrature rule. Numerical examples demonstrate the accuracy of the proposed methods.     

Periodic and solitary wave solutions of the attractive and repulsive nonlinear Schrödinger equation

By means of symbolic computation, the first integral method is presented to obtain novel exact solutions of the nonlinear evolution equation. The obtained results include periodic and solitary wave solutions. The power of this manageable method is confirmed and the availability of symbolic computation is demonstrated.     

Multilevel augmentation methods for nonlinear ill-posed problems

We propose multilevel augmentation methods for solving nonlinear ill-posed problems, involving monotone operators in the Hilbert space by using the Lavrentiev regularization method. This leads to a fast solutions of the discrete regularization methods for the nonlinear ill-posed equations. The regularization parameter choice strategies considered by Pereverzev and Schock (2005) are introduced and the optimal convergence rates of the regularized solutions are obtained. Numerical results are presented to illustrate the accuracy and efficiency of the proposed methods.     

A new semismooth Newton method for NCPs based on the penalized KK function

In this paper, based on the Kanzow-Kleinmichel (KK) function, we introduce a new nonlinear complementarity problem (NCP) function: penalized KK function. We show that the function possesses desirable properties similar to those of the KK function. Based on this new NCP function, we reformulate the NCP to a system of semismooth equations. We also propose a new semismooth Levenberg–Marquardt method to solve the system of semismooth equations that employs both trust region techniques and line searches. The global and quadratic convergence properties can be established under very mild assumptions. Numerical results show the effectiveness of the proposed algorithm and also indicate that superior behaviour of the proposed new NCP function.     

A non-monotone inexact regularized smoothing Newton method for solving nonlinear complementarity problems

In the paper [S.P. Rui and C.X. Xu, A smoothing inexact Newton method for nonlinear complementarity problems, J. Comput. Appl. Math. 233 (2010), pp. 2332–2338], the authors proposed an inexact smoothing Newton method for nonlinear complementarity problems (NCP) with the assumption that F is a uniform P function. In this paper, we present a non-monotone inexact regularized smoothing Newton method for solving the NCP which is based on Fischer–Burmeister smoothing function. We show that the proposed algorithm is globally convergent and has a locally superlinear convergence rate under the weaker condition that F is a P ₀ function and the solution of NCP is non-empty and bounded. Numerical results are also reported for the test problems, which show the effectiveness of the proposed algorithm.     

Metabolic system identification and optimization in continuous culture

To date, there still exist some uncertain factors in the continuous fermentation of glycerol to 1,3-Propanediol (1,3-PD) by Klebsiella pneumoniae because of the limitation in bio-techniques. In this paper, among these uncertain factors, we aim to infer the transport mechanisms of the substrate and the product across the cell membrane of the biomass. On the basis of different inferences of transport mechanisms, we reconstruct various metabolic systems and develop their dynamical systems. To determine the most reasonable metabolic system from all possible ones, we give a quantitative definition of biological robustness and propose an identification model on this basis. An improved Particle Swarm Optimization algorithm is developed to solve the identification model. Numerical results show that the identified system can describe the fermentation process well. Furthermore, to maximize the concentration of 1,3-PD, an optimization model is proposed. Numerical results show that the concentration of 1,3-PD can be increased considerably by employing the obtained optimal strategy.     

New generalized method to construct new non-travelling wave solutions and travelling wave solutions of K–D equations

With the aid of computerized symbolic computation, we obtain new types of general solution of a first-order nonlinear ordinary differential equation with six degrees of freedom and devise a new generalized method and its algorithm, which can be used to construct more new exact solutions of general nonlinear differential equations. The (2+1)-dimensional K–D equation is chosen to illustrate our algorithm such that more families of new exact solutions are obtained, which contain non-travelling wave solutions and travelling wave solutions.     

Pseudo-transient continuation-based variable relaxation solver in nonlinear magnetohydrodynamic simulations

Efficient and robust Variable Relaxation Solver, based on pseudo-transient continuation, is developed to solve nonlinear anisotropic thermal conduction arising from fusion plasma simulations. By adding first- and/or second-order artificial time derivatives to the system, this type of method advances the resulting time-dependent nonlinear PDEs to steady state, which is the solution to be sought. In this process, only the stiffness matrix itself is involved so that the numerical complexity and errors can be greatly reduced. In fact, this work is an extension of integrating efficient linear elliptic solvers for fusion simulation on Cray X1E. Two schemes are derived in this work, first- and second-order variable relaxations. Four factors are observed to be critical for efficiency and preservation of solution's symmetric structure arising from periodic boundary condition: refining meshes in different coordinate directions, initializing nonlinear process, varying time steps in both temporal and spatial directions, and accurately generating nonlinear stiffness matrix. First finer mesh scale should be taken in strong transport direction; next the system is carefully initialized by the solution with linear conductivity; third, time step and relaxation factor are vertex-based varied and optimized at each time step; finally, the nonlinear stiffness matrix is updated by just scaling corresponding linear one with the vector generated from nonlinear thermal conductivity.