On a new numerical algorithm for the solution of triple deck problems

A new algorithm for the solution of laminar viscous-inviscid interactions allowing for the formation of pronounced separation zones is presented. It is then applied to two different types of flows: near critical two layer fluid flow and subsonic flow past an expansion ramp. In the first case emphasis is placed on the formation of so-called non-classical hydraulic jump having the distinguishing property that waves pass through rather than merge with the jump. The second problem exhibits the interesting feature that no solution exists if the ramp angle exceeds a critical value while two solutions exists if the ramp angle is subcritical. The new numerical scheme is used to study the change in the flow behaviour as the critical ramp angle is approached. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constrained mountain pass algorithm for the numerical solution of semilinear elliptic problems

Summary. A new numerical algorithm for solving semilinear elliptic problems is presented. A variational formulation is used and critical points of a C¹-functional subject to a constraint given by a level set of another C¹-functional (or an intersection of such level sets of finitely many functionals) are sought. First, constrained local minima are looked for, then constrained mountain pass points. The approach is based on the deformation lemma and the mountain pass theorem in a constrained setting. Several examples are given showing new numerical solutions in various applications.Mathematics Subject Classification (2000):35J20, 65N99The author would like to thank the referee for helpful comments in particular on Section 4.     

A generalized structured doubling algorithm for the numerical solution of linear quadratic optimal control problems

We propose a generalization of the structured doubling algorithm to compute invariant subspaces of structured matrix pencils that arise in the context of solving linear quadratic optimal control problems. The new algorithm is designed to attain better accuracy when the classical Riccati equation approach for the solution of the optimal control problem is not well suited because the stable and unstable invariant subspaces are not well separated (because of eigenvalues near or on the imaginary axis) or in the case when the Riccati solution does not exist at all. We analyze the convergence of the method and compare the new method with the classical structured doubling algorithm as well as some structured QR methods. Copyright © 2012 John Wiley & Sons, Ltd.     

A numerical algorithm for nonlinear multi-point boundary value problems

In this paper, an algorithm is presented for solving second-order nonlinear multi-point boundary value problems (BVPs). The method is based on an iterative technique and the reproducing kernel method (RKM). Two numerical examples are provided to show the reliability and efficiency of the present method.     

A sharp error estimate for the numerical solution of multivariate Dirichlet problems

For the multidimensional Dirichlet problem of the Poisson equation on an arbitrary compact domain, this study examines convergence properties with rates of approximate solutions, obtained by a standard difference scheme over inscribed uniform grids. Sharp quantitative estimates are given by the use of second moduli of continuity of the second single partial derivatives of the exact solution. This is achieved by employing the probabilistic method of simple random walk.     

A new PEC algorithm for the numerical solution of ordinary differential equations

Significant time reduction in obtaining numerical solutions of ordinary differential equations for which function evaluations are time consuming can be obtained with PEC methods as compared to PECE methods. In this report we present two PEC methods: a fourth-order algorithm for which stability characteristics and numerical examples are presented, and a second-order algorithm which is just mentioned. It is believed that PEC methods represent a useful addition to the library of solution techniques.     

A numerical algorithm for some singularly perturbed boundary value problems

A numerical algorithm is proposed to solve singularly perturbed linear two-point value problems. The method starts with a partial decoupling of the system to obtain two independent subsystems, fast and slow components. Each subsystem is then solved separately. A second-order finite difference scheme is used for this purpose. Numerical examples will be presented to show the efficiency of the method.     

A Taylor series method for the numerical solution of two-point boundary value problems

Summary For the numerical solution of two-point boundary value problems a shooting algorithm based on a Taylor series method is developed. Series coefficients are generated automatically by recurrence formulas. The performance of the algorithm is demonstrated by solving six problems arising in nonlinear shell theory, chemistry and superconductivity.     

A boundary element method for Signorini problems in three dimensions

Summary In this paper, a Signorini problem in three dimensions is reduced to a variational inequality on the boundary, and a boundary element method is described for the numerical approximation of its solution; an optimal error estimate is also given.This work is supported in part by the National Natural Science Foundation of China, and by the Royal Society of London     

A high order method for the numerical solution of two-point boundary value problems

A one step finite difference scheme of order 4 for the numerical solution of the general two-point boundary value problemy=f(t,y),a t b, withg(y(a),y(b))=0 is presented. The global discretization error of the scheme is shown, in sufficiently smooth cases, to have an asymptotic expansion containing even powers of the mesh size only. This justifies the use of Richardson extrapolation (or deferred correction) to obtain high orders of accuracy. A theoretical examination of the new scheme for large systems of equations shows that for a given mesh size it generally requires about twice as much work as the Keller box scheme. However, the expectation of higher accuracy usually justifies this extra computational effort. Some numerical results are given which confirm these expectations and show that the new scheme can be generally competitive with the box scheme.     

A numerical algorithm for the solution of an intermediate fractional advection dispersion equation

One of the main applications of fractional derivative is in the modeling of intermediate physical processes. In this work, the methodology of fractional calculus is used to model the intermediate process between advection and dispersion as an initial-boundary-value problem for a partial differential equation of fractional order with one spatial variable and constant coefficients. A numerical algorithm based on symbolic computations for the solution of the problem is suggested and tested with good results.     

The method of fundamental solutions for Signorini problems

We investigate the use of method of fundamental solutions (MFS)for the numerical solution of Signorini boundary value problems.The MFS is an ideal candidate for solving such problems becauseinequality conditions alternating at unknown points of the boundarycan be incorporated naturally into the least-squares minimizationscheme associated with the MFS. To demonstrate its efficiency,we apply the method to two Signorini problems. The first isa groundwater flow problem related to percolation in gentlysloping beaches, and the second is an electropainting application.For both problems, the results are in close agreement with previouslyreported numerical solutions.     

The RKGL method for the numerical solution of initial-value problems

We introduce the RKGL method for the numerical solution of initial-value problems of the form y^′=f(x,y)$y^{\prime }=f(x,y)$, y(a)=α$y(a)=\alpha$. The method is a straightforward modification of a classical explicit Runge–Kutta (RK) method, into which Gauss–Legendre (GL) quadrature has been incorporated. The idea is to enhance the efficiency of the method by reducing the number of times the derivative f(x,y)$f(x,y)$ needs to be computed. The incorporation of GL quadrature serves to enhance the global order of the method by, relative to the underlying RK method. Indeed, the RKGL method has a global error of the form Ah^r+1+Bh^2m${\mathit{Ah}}^{r+1}+{\mathit{Bh}}^{2m}$, where r is the order of the RK method and m is the number of nodes used in the GL component. In this paper we derive this error expression and show that RKGL is consistent, convergent and strongly stable.     

Approximate method for the numerical solution of singular perturbation problems

We present an approximate method for the numerical solution of linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. It is motivated by the asymptotic behavior of singular perturbation problems. The original problem is divided into inner and outer region problems. The reduced problem is solved to obtain the terminal boundary condition. Then, a new inner region problem is created and solved as a two point boundary value problem. In turn, the outer region problem is also modified and the resulting problem is efficiently treated by employing the trapezoidal formula coupled with discrete invariant imbedding algorithm. The proposed method is iterative on the terminal point. Some numerical experiments have been included to demonstrate its applicability.     

Potential-based numerical solution of Dirichlet problems for the Helmholtz equation

Three-dimensional Dirichlet problems for the Helmholtz equation are considered in generalized formulations. By applying single-layer potentials, they are reduced to Fredholm boundary integral equations of the first kind. The equations are discretized using a special averaging method for integral operators with weak singularities in the kernels. As a result, the integral equations are approximated by systems of linear algebraic equations with easy-to-compute coefficients, which are solved numerically by applying the generalized minimal residual method. A modification of the method is proposed that yields solutions in the spectra of interior Dirichlet problems and integral operators when the integral equations are not equivalent to the original differential problems and are not well-posed. Numerical results are presented for assessing the capabilities of the approach.     

A scenario aggregation algorithm for the solution of stochastic dynamic minimax problems

In this paper, we present a scenario aggregation algorithm for the solution of the dynamic minimax problem in stochastic programming. We consider the case where the joint probability distribution has a known finite support. The algorithm applies the Alternating Direction of Multipliers Method on a reformulation of the minimax problem using a double duality framework. The problem is solved by decomposition into scenario sub-problems, which are deterministic multi-period problems. Convergence properties are deduced from the Alternating Direction of Multipliers. The resulting algorithm can be seen as an extension of Rockafellar and Wets Progressive Hedging algorithm to the dynamic minimax context.