An asymptotic-numerical hybrid method for singularly perturbed system of two-point reaction-diffusion boundary-value problems

This article focuses on the numerical approximate solution of singularly perturbed systems of secondorder reaction-diffusion two-point boundary-value problems for ordinary differential equations. To handle these types of problems, a numerical-asymptotic hybrid method has been used. In this hybrid approach, an efficient asymptotic method, the so-called successive complementary expansion method (SCEM) is employed first, and then a numerical method based on finite differences is applied to approximate the solution of corresponding singularly perturbed reactiondiffusion systems. Two illustrative examples are provided to demonstrate the efficiency, robustness, and easy applicability of the present method with convergence propertiesNo sponso

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters

Preprint version, the final publication is available at Springer via http://dx.doi.org/10.1007/s10543-015-0559-8This paper discusses the numerical solution of 1-D convection-diffusion-reaction problems that are singularly perturbed with two small parameters using a new mesh-adaptive upwind scheme that adapts to the boundary layers. The meshes are generated by the equidistribution of a special positive monitor function. Uniform, parameter independent convergence is shown and holds even in the limit that the small parameters are zero. Numerical experiments are presented that illustrate the theoretical findings, and show that the new approach has better accuracy compared with current methods.DFG, SFB 1029, Substantial efficiency increase in gas turbines through direct use of coupled unsteady combustion and flow dynamic

Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling

In the present article we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with appropriate predictor-corrector method to achieve higher resolution. The underlying finite volume scheme is conservative and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed, thus unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.Comment: 46 pages, 7 figures, 7 tables, 94 references. Accepted to Geosciences. Other author's papers can be downloaded at http://www.denys-dutykh.com