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During this period research was continued on the development and application of numerical methods for singularly-perturbed (or stiff) boundary value problems for ordinary differential equations and initial-boundary value problems for partial differential equations. The author concentrated most heavily on extensions to the adaptive finite element methods for partial differential equations. In particular, the stability of several mesh moving schemes was analyzed and local refinement techniques developed. The author also has some encouraging preliminary results on mesh moving methods in two dimensions. The investigators are applying their methods to several interesting physical problems, such as elastic-plastic solids, combustion, and a nonlinear Schrodinger equation which exhibits self focusing. (Author)

This paper focuses on solving singularly perturbed parabolic equations of the convection-diffusion type with a large negative spatial shift and an integral boundary condition. A higher-order uniformly convergent numer-ical approach is proposed that uses Crank–Nicolson and a hybrid finite difference approximation on a piece-wise uniform Shishkin mesh. Simp-son’s 1/3 integration rule is used to treat the integral boundary condition. The proposed method has been shown to achieve almost second-order uni-form convergence. The computational results derived from the numerical experiment are consistent with the theoretical estimates. Furthermore, the method produces a more accurate result than certain other methods in the literature.

In this article, singularly perturbed differential difference equations having delay and advance in the reaction terms are considered. The highest-order derivative term of the equation is multiplied by a perturbation parameter ε taking arbitrary values in the interval (0 , 1] . For the small value of ε , the solution of the equation exhibits a boundary layer on the left or right side of the domain depending on the sign of the convective term. The terms with the shifts are approximated by using the Taylor series approximation.The resulting singularly perturbed boundary value problem is solved using an exponentially fitted tension spline method. The stability and uniform convergence of the scheme are discussed and proved. Numerical exam ples are considered for validating the theoretical analysis of the scheme. The developed scheme gives an accurate result with linear order uniform convergence.

We construct analytic and formal solutions for a family of $q$-difference-differential problems, under the action of a perturbation parameter. This work is a continuation of a previous work focusing on a singularly perturbed $q$-difference-differential problem for which a phenomenon of multilevel $q$-Gevrey asymptotics has been observed, owing to the fact that the main equation is factorized as a product of two simpler equations in so-called normal forms, each producing one single level of $q$-Gevrey asymptotics. The problem under study in this paper is a priori not factorizable. We follow instead a direct approach developed in a contribution of the first author based on successive applications of two $q$-Borel-Laplace transforms of different orders both to the same initial problem and which can be described by means of a Newton polygon.

This document described the development and application of numerical methods for singularity perturbed two-point boundary value problems and time-dependent partial differential equations. (Author)

We study the asymptotic behavior of the solutions related to a family of singularly perturbed partial differential equations in the complex domain. The analytic solutions are asymptotically represented by a formal power series in the perturbation parameter. The geometry of the problem and the nature of the elements involved in it give rise to different asymptotic levels related to the so-called strongly regular sequences. The result leans on a novel version of a multi-level Ramis-Sibuya theorem.

We introduce a micro-macro parareal algorithm for the time-parallel integration of multiscale-in-time systems. The algorithm first computes a cheap, but inaccurate, solution using a coarse propagator (simulating an approximate slow macroscopic model), which is iteratively corrected using a fine-scale propagator (accurately simulating the full microscopic dynamics). This correction is done in parallel over many subintervals, thereby reducing the wall-clock time needed to obtain the solution, compared to the integration of the full microscopic model. We provide a numerical analysis of the algorithm for a prototypical example of a micro-macro model, namely singularly perturbed ordinary differential equations. We show that the computed solution converges to the full microscopic solution (when the parareal iterations proceed) only if special care is taken during the coupling of the microscopic and macroscopic levels of description. The convergence rate depends on the modeling error of the approximate macroscopic model. We illustrate these results with numerical experiments.

During the period covered by this report, the investigator continued his research on the development and application of numerical methods for singularly-perturbed (or stiff) boundary value problems for ordinary differential equations. Results were obtained for collocation and finite difference methods for scalar and vector systems of two-point boundary value problems and for adaptive grid finite element methods for partial differential equations. His exponentially weighted finite element and spline in tension methods are being applied to partial differential equations as well as to ordinary differential equations. (Author)

Research was continued on the development and applications of adaptive numerical methods for singularly perturbed initial-boundary value problems for partial differential equations. Analysis is made of mesh moving schemes, examined local refinement methods, and developed a posteriori error estimation technique for none-and-two-dimensional hyperbolic and parabolic problems. Development has begun on parallel versions of some adaptive procedures. These methods are applied to several interesting physical problems, that arise in, e.g., elastic-plastic deformation, combustion, and fluid mechanics.

Research was continued on the development and applications of adaptive numerical methods for singularly perturbed initial-boundary value problems for partial differential equations. Analysis is made of mesh moving schemes, examined local refinement methods, and developed a posteriori error estimation technique for none-and-two-dimensional hyperbolic and parabolic problems. Development has begun on parallel versions of some adaptive procedures. These methods are applied to several interesting physical problems, that arise in, e.g., elastic-plastic deformation, combustion, and fluid mechanics.

Research was continued on the development and applications of adaptive numerical methods for singularly perturbed initial-boundary value problems for partial differential equations. Analysis is made of mesh moving schemes, examined local refinement methods, and developed a posteriori error estimation technique for none-and-two-dimensional hyperbolic and parabolic problems. Development has begun on parallel versions of some adaptive procedures. These methods are applied to several interesting physical problems, that arise in, e.g., elastic-plastic deformation, combustion, and fluid mechanics.

This article deals with an effcient approximation method named successive complementary expansion method (SCEM) for solving singularly perturbed differential-difference equations with mixed shifts. It is compared with the method of matched asymptotic expansion (MMAE) and the parameter uniform upwind finite difference scheme for solving such a model. The comparison shows, unlike the MMAE, the SCEM method requires no matching procedure. It requires less computation when compared to the upwind finite difference scheme on the Shishkin mesh. The error analysis is carried out to prove the robustness of the method. Some numerical experiments are provided, which show the effectiveness of the proposed method.

During the period covered by this grant, we conducted research on the development and application of adaptive numerical methods for singularly perturbed initial-boundary value problems for partial differential equations. We studied both local refinement methods and methods of lines techniques using properly nested and overlapping grids. The geometric modeling capabilities of our procedures were improved through the use of finite quadtree and octree encoding schemes; parallel versions of adaptive procedures on shared-memory computers were developed; and variable-order finite element methods to be used with adaptive p- and hp-refinement were created.

During the three-year period of this project, we conducted research on the development, analysis, and application of serial and parallel adaptive computational strategies for solving transient and steady partial differential systems. Concentrating on high-order methods and adaptive approaches that combine mesh refinement and coarsening (h-refinement, order variation (p- refinement), and occasionally, mesh motion (r-refinement), we addressed problems in combustion, materials science and compressible fluid mechanics. Special spatially-discrete finite element Galerkin methods were considered for the parallel and adaptive solution of hyperbolic conservation laws. Improved solution-limiting and error-estimation strategies increased the accuracy and efficiency of these methods which are being applied to two - and three- dimensional compressible flow problems. Adaptive techniques for dissipative systems are being applied to problems in the manufacture of ceramic composite media.

Abstracts

During the period covered by this report we continued our research on the development and application of numerical methods for singularly-perturbed (or stiff) boundary value problems for ordinary differential equations and initial-boundary value problems for partial differential equations. Results were obtained for collocation methods for vector systems of two-point boundary value problems and for adaptive grid finite element methods for partial differential equations. We are applying our methods to several interesting physical problems, such as, the deformation of nonlinear elastic beams and a nonlinear Schrodinger equation which exhibits self focusing. (Author)

During this one-year project, we continued our research on the development, analysis, and application of serial and parallel adaptive computational strategies for solving transient and steady partial differential systems. We concentrated on high-order methods and adaptive approaches that unite mesh refinement and coarsening (h-refinement), order variation (p- refinement), and mesh motion (r-refinement). Parallel computational techniques involved load-balancing and load-redistribution strategies for implementing these adaptive methods on distributed-memory MIMD computers. In particular, we have developed migration strategies that exchange finite elements between neighboring spatial domains of different processors. Effective load balancing in an adaptive setting requires speedy procedures since balancing must be performed frequently. Migration offers several advantages in this regard since it (i) has a low unit cost, (ii) can take advantage of locality, and (iii) can improve communications volumes. Procedures tested in two dimensional situations are being extended to three dimensions and preliminary methods, Singularly perturbed equations, Partial differential equations, Parallel computation.

During the course of this grant algorithms were developed and analyzed for the numerical solution of singularly-perturbed (or stiff) initial value and boundary value problems for ordinary differential equations and initial-boundary value problems for partial differential equations. These general purpose methods have been applied to a wide variety of problems arising in several disciplines, including optimization and optimal control, nonlinear oscillations, chemical reactions, and hydrodynamic stability. Keywords: Computations, Plane poiseville flow, Viscous flow.

We study the asymptotic behavior of the solutions related to a family of singularly perturbed linear partial differential equations in the complex domain. The analytic solutions obtained by means of a Borel-Laplace summation procedure are represented by a formal power series in the perturbation parameter. Indeed, the geometry of the problem gives rise to a decomposition of the formal and analytic solutions so that a multi-level Gevrey order phenomenon appears. This result leans on a Malgrange-Sibuya theorem in several Gevrey levels.

During the period covered by this report the investigators continued their research on the development and application of adaptive numerical methods for singularly perturbed initial-boundary value problems for partial differential equations. They continued their analysis of the stability of mesh moving schemes for one-dimensional parabolic problems. Their also developed at moving mesh scheme with local refinement for two-dimensional hyperbolic systems and are considering a similar scheme for parabolic problems. They are applying our methods to several interesting physical problems, such as, elastic-plastic solids, combustion, and a nonlinear Schrodinger equation which exhibits self-focusing. (Author)

A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin piecewise-uniform meshes are introduced, which are used in conjunction with a classical finite difference discretisation, to construct two numerical methods for solving this problem. It is proved that the numerical approximations obtained with these methods are essentially first, respectively second, order convergent uniformly with respect to all of the parameters.

During the period covered by this report we continued our research on the development and application of adaptive numerical methods for singularly perturbed initial-boundary value problems for partial differential equations. We continued our analysis of the stability of mesh moving schemes and developed local refinement and moving mesh schemes for one- and two-dimensional hyperbolic and parabolic problems. We are applying our methods to several interesting physical problems, such as, elastic-plastic solids, combustion, and a nonlinear Schrodinger equation which exhibits self-focusing. Keywords: Bibliographies; Abstracts.

In this paper, singularly perturbed nonlinear differential/algebraic equations (DAEs) are considered and a proof of the existence and uniqueness of a solution is given. Asymptotic expansions for such a solution are obtained and proved to be uniformly convergent. This generalizes known results about asymptotic expansions of singularly perturbed ordinary differential equations.