# Parameter-Uniform Numerical Methods for a Class of Parameterized Singular Perturbation Problems

TL;DR: In this paper, a weighted finite difference scheme is proposed for solving a class of parameterized singularly perturbed problems (SPPs). Depending upon the choice of the weight parameter, the scheme is automatically transformed fromthe backward Euler scheme to amonotone hybrid scheme.

Abstract: In this article, a weighted finite difference scheme is proposed for solving a class of parameterized singularly perturbed problems (SPPs). Depending upon the choice of the weight parameter, the scheme is automatically transformed fromthe backward Euler scheme to amonotone hybrid scheme. Three kinds of nonuniform grids are considered, especially the standard Shishkin mesh, the Bakhavalov–Shishkinmesh and the adaptive grid. Themethods are shown to be uniformly convergent with respect to the perturbation parameter for all three types of meshes. The rate of convergence is of first order for the backward Euler scheme and second order for themonotone hybrid scheme. Furthermore, the proposed method is extended to a parameterized problem with mixed type boundary conditions and is shown to be uniformly convergent. Numerical experiments are carried out to show the efficiency of the proposed schemes, which indicate that the estimates are optimal.

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06 Sep 2021

TL;DR: This work considers methods that can be applied for building cyber-physical systems and digital twins and the general neural network approach as a variant of machine learning modeling with physics-based regularization (or physics-informed neural networks) and their combination.

Abstract: This work is devoted to the description and comparative study of some methods of mathematical modeling. We consider methods that can be applied for building cyber-physical systems and digital twins. These application areas add to the usual accuracy requirements for a model the need to be adaptable to new data and the small computational complexity allows it to be used in embedded systems. First, we regard the finite element method as one of the “pure” physics-based modeling methods and the general neural network approach as a variant of machine learning modeling with physics-based regularization (or physics-informed neural networks) and their combination. A physics-based network architecture model class has been developed by us on the basis of a modification of classical numerical methods for solving ordinary differential equations. The model problem has a parameter at some values for which the phenomenon of stiffness is observed. We consider a fixed parameter value problem statement and a case when a parameter is one of the input variables. Thus, we obtain a solution for a set of parameter values. The resulting model allows predicting the behavior of an object when its parameters change and identifying its parameters based on observational data.

4 citations

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TL;DR: Numerical experiments show that the improved a posteriori error estimation for the difference scheme on an arbitrary mesh is second-order uniformly convergent and improves previous results.

2 citations

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TL;DR: In this article, a finite difference scheme of hybrid type with an appropriate Shishkin mesh is proposed to solve the problem, which is of almost second order convergent in the discrete maximum norm.

2 citations

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TL;DR: In this article , a finite difference scheme of hybrid type with an appropriate Shishkin mesh is proposed to solve the problem, which is of almost second order convergent in the discrete maximum norm.

2 citations

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TL;DR: In this article, a coupled system of singularly perturbed first order differential equations with discontinuous source term having prescribed initial conditions is examined and a backward difference scheme is applied on the layer adapted meshes and a uniform convergence is established with respect to all parameters.

Abstract: We examine a coupled system of $$m( \ge 2)$$
singularly perturbed first order differential equations with discontinuous source term having prescribed initial conditions. The small perturbation parameters of different magnitudes are associated with the leading term of each equation. In general, the system does not satisfy the standard maximum principle. The solution exhibits the initial and interior layers which overlap and interact. A piecewise-uniform Shishkin mesh and graded Bakhvalov mesh are constructed for the considered problem. A backward difference scheme is applied on the layer adapted meshes and a uniform convergence is established with respect to all parameters. Numerical experiments are described for both the Shishkin and Bakhvalov meshes and comparison is done with an approximated scheme on the uniform mesh.

##### References

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30 Mar 2000

TL;DR: In this paper, numerical methods for problems with Boundary Layers are presented. But they do not address the problems with Frictionless Walls and No Slip Boundary Conditions, and they are not suitable for Non-Monotone Methods in two dimensions.

Abstract: Introduction to Numerical Methods for Problems with Boundary Layers Numerical Methods on Uniform Meshes Layer Resolving Methods for Convection-Diffusion Problems in One Dimension The Limitations of Non-Monotone Numerical Methods Convection-Diffusion Problems in a Moving Medium Convection-Diffusion Problems with Frictionless Walls Convection-Diffusion Problems with No Slip Boundary Conditions Experimental Estimation of Errors Non-Monotone Methods in Two Dimensions Linear and Nonlinear Reaction-Diffusion Problems Prandtl Flow past a Flat Plate-Blasius' Method Prandtl Flow past a Flat Plate-Direct Method References.

725 citations

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TL;DR: Some three point difference schemes are considered for a singular perturbation problem without turning points in this article, and bounds for the discretization error are obtained which are uniformly valid for all h and e > 0.

Abstract: Some three point difference schemes are considered for a singular perturbation problem without turning points Bounds for the discretization error are obtained which are uniformly valid for all h and e > 0 The degeneration of the order of the schemes at e = 0 is considered

408 citations

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12 Mar 2014

TL;DR: Applications of geometric methods to the analysis of numerical grid behavior as well as grid generation based on the minimization of functionals of smoothness, conformality, orthogonality, energy, and alignment complete the second edition of this outstanding compendium on grid generation methods.

Abstract: This book is an introduction to structured and unstructured grid methods in scientific computing, addressing graduate students, scientists as well as practitioners. Basic local and integral grid quality measures are formulated and new approaches to mesh generation are reviewed. In addition to the content of the successful first edition, a more detailed and practice oriented description of monitor metrics in Beltrami and diffusion equations is given for generating adaptive numerical grids. Also, new techniques developed by the author are presented, in particular a technique based on the inverted form of Beltramis partial differential equations with respect to control metrics. This technique allows the generation of adaptive grids for a wide variety of computational physics problems, including grid clustering to given function values and gradients, grid alignment with given vector fields, and combinations thereof. Applications of geometric methods to the analysis of numerical grid behavior as well as grid generation based on the minimization of functionals of smoothness, conformality, orthogonality, energy, and alignment complete the second edition of this outstanding compendium on grid generation methods.

318 citations

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01 Jan 1974

TL;DR: In this paper, the authors consider approximation of a given function f, on [0, 1] say, by elements of S π k, i.e., by polynomial splines of order k (or, degree < k) on some partition.

Abstract: Consider approximation of a given function f, on [0,1] say, by elements of S π k , i.e., by polynomial splines of order k (or, degree < k) on some partition
$$({t_i})_0^{N + 1}of[0,1]\;0 = {t_0} < {t_1} \leqslant {t_2} \leqslant ... \leqslant {t_N} < {t_{n + 1}} = 1$$
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311 citations