This paper proposes a numerical method based on the Adomian decomposition approach for the time discretization, applied to Euler equations. A recursive property is demonstrated that allows to formulate the method in an appropriate and efficient way. To obtain a fully numerical scheme, the space discretization is achieved using the classical DG techniques. The efficiency of the obtained numerical scheme is demonstrated through numerical tests by comparison to exact solution and the popular Runge-Kutta DG method results.
Application of Differential Transform Method for El Nino Southern Oscillation (ENSO) Model with compared Adomian Decomposition and Variational Iteration Methods
Kamil Ozdag Science Faculty of Karamanoglu Mehmetbey University（卡拉曼奥萨格科学学院，卡拉曼奥格鲁梅特贝大学）；Science Faculty of Selcuk University（塞琉古大学理学院）
Some Solutions of Fractional Order Partial Differential Equations Using Adomian Decomposition Method
University of Management and Technology, Lahore, Pakistan（拉合尔管理技术大学）；The University of Lahore, Lahore, Pakistan（拉合尔大学）；University of Science and Technology of China, China（中国科学技术大学）
This article aims to establish a semi-analytical approach based on the homotopy perturbation method (HPM) to find the closed form or approximated solutions for the population balance equations such as Smoluchowski's coagulation, fragmentation, coupled coagulation-fragmentation and bivariate coagulation equations. An accelerated form of the HPM is combined with the Elzaki transformation to improve the accuracy and efficiency of the method. One of the significant advantages of the technique lies over the classic numerical methods as it allows solving the linear and non-linear differential equati…
The Smoluchowski's aggregation equation has applications in the field of bio-pharmaceuticals \cite{zidar2018characterisation}, financial sector \cite{PUSHKIN2004571}, aerosol science \cite{shen2020efficient} and many others. Several analytical, numerical and semi-analytical approaches have been devised to calculate the solutions of this equation. Semi-analytical methods are commonly employed since they do not require discretization of the space variable. The article deals with the introduction of a novel semi-analytical technique called the optimized decomposition method (ODM) (see \cite{odiba…
We study a natural alternating method of Schwarz type (domain decomposition) for certain class of couplings between local and nonlocal operators. We show that our method fits into Lion's framework and prove, as a consequence, convergence in both, the continuous and the discrete settings.
The error analysis of a proper orthogonal decomposition (POD) data assimilation (DA) scheme for the Navier-Stokes equations is carried out. A grad-div stabilization term is added to the formulation of the POD method. Error bounds with constants independent on inverse powers of the viscosity parameter are derived for the POD algorithm. No upper bounds in the nudging parameter of the data assimilation method are required. Numerical experiments show that, for large values of the nudging parameter, the proposed method rapidly converges to the real solution, and greatly improves the overall accurac…
Since proposed in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229: 3091--3120, 2010], the Zhang--Shu framework has attracted extensive attention and motivated many bound-preserving (BP) high-order discontinuous Galerkin and finite volume schemes for various hyperbolic equations. A key ingredient in the framework is the decomposition of the cell averages of the numerical solution into a convex combination of the solution values at certain quadrature points, which helps to rewrite high-order schemes as convex combinations of formally first-order schemes. The classic convex decomposition originall…
The reduction of computational costs in the numerical solution of nonstationary problems is achieved through splitting schemes. In this case, solving a set of less computationally complex problems provides the transition to a new level in time. The traditional construction approach of splitting schemes is based on an additive representation of the problem operator(s) and uses explicit-implicit approximations for individual terms. Recently (Y. Efendiev, P.N. Vabishchevich. Splitting methods for solution decomposition in nonstationary problems. \textit{Applied Mathematics and Computation}. \text…
Stability and convergence analysis for the domain decomposition finite element/finite difference (FE/FD) method is presented. The analysis is designed for semi-discrete finite element scheme for the time-dependent Maxwell's equations. The explicit finite element schemes in different settings of the spatial domain are constructed and domain decomposition algorithm is formulated. Several numerical examples validate convergence rates obtained in the theoretical studies.
In this paper, we present a novel nodal integration scheme for meshfree Galerkin methods that draws on the mathematical framework of the virtual element method. We adopt linear maximum-entropy basis functions for the discretization of field variables, although the proposed scheme is applicable to any linear meshfree approximant. In our approach, the weak form integrals are nodally integrated using nodal representative cells that carry the nodal displacements and state variables such as strains and stresses. The nodal integration is performed using the virtual element decomposition, wherein the…
In this work, we consider compressible single-phase flow problems in a porous media containing a fracture. In the latter, a non-linear pressure-velocity relation is prescribed. Using a non-overlapping domain decomposition procedure, we reformulate the global problem into a non-linear interface problem. We then introduce two new algorithms that are able to efficiently handle the non-linearity and the coupling between the fracture and the matrix, both based on linearization by the so-called L-scheme. The first algorithm, named MoLDD, uses the L-scheme to resolve the non-linearity, requiring at e…
In this paper, we propose two novel Robin-type domain decomposition methods based on the two-grid techniques for the coupled Stokes-Darcy system. Our schemes firstly adopt the existing Robin-type domain decomposition algorithm to obtain the coarse grid approximate solutions. Then two modified domain decomposition methods are further constructed on the fine grid by utilizing the framework of two-grid methods to enhance computational efficiency, via replacing some interface terms by the coarse grid information. The natural idea of using the two-grid frame to optimize the domain decomposition met…
The aim of this paper is to analyze from a mathematical perspective some existing schemes to partition a molecular density into several atomic contributions, with a specific focus on Iterative Stockholder Atom (ISA) methods. We provide a unified mathematical framework to describe the latter family of methods and propose a new scheme, named L-ISA (for linear approximation of ISA). We prove several important mathematical properties of the ISA and L-ISA minimization problems and show that the so-called ISA algorithms can be viewed as alternating minimization schemes, which in turn enables us to o…
A novel high-order numerical scheme is proposed to compute the covariant derivative, particularly for divergence and curl, on any curved surface. The proposed scheme does not require the construction of a curved axis or metric tensor, which would deteriorate the accuracy of the covariant derivative and prevent its application to complex surfaces. As an application, the Helmholtz-Hodge decomposition (HHD) is adapted in the context of the Galerkin method for displaying the irrotational, incompressible, and harmonic components of vectors on curved surfaces.
A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling
INRIA Rennes - Bretagne Atlantique（法国国家信息与自动化研究所雷恩-大西洋布列塔尼中心）；Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, Université de Nantes（南特大学让·勒雷数学实验室，CNRS UMR 6629）；Institut de Recherche Mathématiques de Rennes, CNRS UMR 6625, Université de Rennes 1（雷恩大学1号数学研究中心，CNRS UMR 6625）；ENS Rennes（雷恩高等师范学院）
Cauchy problem for an abstract hyperbolic equation with the Lipschitz continuous operator is considered in the Hilbert space. The operator corresponding to the elliptic part of the equation is a sum of operators $A_{1},\,A_{2},\,\ldots,\,A_{m}$. Each addend is a self-adjoint and positive definite operator. A parallel type decomposition scheme for an approximate solution of the stated problem is constructed. The main idea of the scheme is that on each local interval classic difference problems are solved in parallel (independently from each other) respectively with the operators $A_{1},\,A_{2},…
We propose novel less diffusive schemes for conservative one- and two-dimensional hyperbolic systems of nonlinear partial differential equations (PDEs). The main challenges in the development of accurate and robust numerical methods for the studied systems come from the complicated wave structures, such as shocks, rarefactions and contact discontinuities, arising even for smooth initial conditions. In order to reduce the diffusion in the original central-upwind schemes, we use a local characteristic decomposition procedure to develop a new class of central-upwind schemes. We apply the develope…
Compact finite-difference (FD) schemes specify derivative approximations implicitly, thus to achieve parallelism with domain-decomposition suitable partitioning of linear systems is required. Consistent order of accuracy, dispersion, and dissipation is crucial to maintain in wave propagation problems such that deformation of the associated spectra of the discretized problems is not too severe. In this work we consider numerically tuning spectral error, at fixed formal order of accuracy to automatically devise new compact FD schemes. Grid convergence tests indicate error reduction of at least a…
When constructing high-order schemes for solving hyperbolic conservation laws, the corresponding high-order reconstructions are commonly performed in characteristic spaces to eliminate spurious oscillations as much as possible. For multi-dimensional finite volume (FV) schemes, we need to perform the characteristic decomposition several times in different normal directions of the target cell, which is very time-consuming. In this paper, we propose a rotated characteristic decomposition technique which requires only one-time decomposition for multi-dimensional reconstructions. The rotated direct…
