Compact Finite Research Articles

A high-order coupled compact integrated RBF approximation based domain decomposition algorithm for second-order differential problems

This paper presents a high-order coupled compact integrated RBF (CCIRBF) approximation based domain decomposition (DD) algorithm for the discretisation of second-order differential problems. Several Schwarz DD algorithms, including one-level additive/ multiplicative and two-level additive/ multiplicative/ hybrid, are employed. The CCIRBF based DD algorithms are analysed with different mesh sizes, numbers of subdomains and overlap sizes for Poisson problems. Our convergence analysis shows that the CCIRBF two-level multiplicative version is the most effective algorithm among various schemes employed here. Especially, the present CCIRBF two-level method converges quite rapidly even when the domain is divided into many subdomains, which shows great promise for either serial or parallel computing. For practical tests, we then incorporate the CCIRBF into serial and parallel two-level multiplicative Schwarz. Several numerical examples, including those governed by Poisson and Navier-Stokes equations are analysed to demonstrate the accuracy and efficiency of the serial and parallel algorithms implemented with the CCIRBF. Numerical results show: (i) the CCIRBF-Serial and -Parallel algorithms have the capability to reach almost the same solution accuracy level of the CCIRBF-Single domain, which is ideal in terms of computational calculations; (ii) the CCIRBF-Serial and -Parallel algorithms are highly accurate in comparison with standard finite difference, compact finite difference and some other schemes; (iii) the proposed CCIRBF-Serial and -Parallel algorithms may be used as alternatives to solve large-size problems which the CCIRBF-Single domain may not be able to deal with. The ability of producing stable and highly accurate results of the proposed serial and parallel schemes is believed to be the contribution of the coarse mesh of the two-level domain decomposition and the CCIRBF approximation. It is noted that the focus of this paper is on the derivation of highly accurate serial and parallel algorithms for second-order differential problems. The scope of this work does not cover a thorough analysis of computational time.

A three-point coupled compact integrated RBF scheme for second-order differential problems

In this paper, we propose a three-point coupled compact integrated radial basis function (CCIRBF) approximation scheme for the discretisation of second-order differential problems in one and two dimensions. The CCIRBF employs integrated radial basis functions (IRBFs) to construct the approximations for its first and second derivatives over a three-point stencil in each direction. Nodal values of the first and second derivatives (i.e. extra information), incorporated into approximations by means of the constants of integration, are simultaneously employed to compute the first and second derivatives. The essence of the CCIRBF scheme is to couple the extra information of the nodal first and second derivative values via their identity equations. Owing to its coupling of the information of the nodal first and second derivatives, the CCIRBF scheme becomes more accurate, stable and efficient than the normal compact integrated radial basis function (CIRBF) schemes proposed by [Thai-Quang, N.; Mai-Duy, N.; Tran, C.-D.; Tran-Cong, T. (2012)]. The main features of the CCIRBF scheme include: three-point, high-order accuracy, stability, efficiency and inclusion of boundary values. Several analytic problems are considered to verify the present scheme and to compare its accuracy, stability and efficiency with those of the CIRBF, higher-order compact finite difference (HOC) and some other high-order schemes. Numerical results show that highly accurate and stable results are obtained with the proposed scheme. Additionally, the present scheme also takes less time to achieve target accuracy in comparison with the CIRBF and HOC schemes.

On behavior of preconditioned methods for a class of compact finite difference schemes in solution of hyperbolic equations

In this article, for a class of linear systems arising from the compact finite difference schemes, we apply Krylov subspace methods in combination the ADI, BLAGE,... preconditioners. We consider our scheme in solution of hyperbolic equations subject to appropriate initial and Dirichlet boundary conditions, where is constant. We show, the BLAGE preconditioner is extremely effective in achieving optimal convergence rates. Numerical results performed on model problems to confirm the efficiency of our approach.

Compact Hypergroup Characters and Finite Universal Korovkin Sets in L 1-Hypergroup Algebra

Let K be a compact hypergroup.We investigate Trig(K), the linear span of coordinate functions of the irreducible representations of K. Contrary to the group case, Trig(K) endowed with the usual multiplication does not bear an algebra structure, but it has a natural normed algebra structure when it inherits the convolution from $$\mathcal {M}(K)$$ , the algebra of all bounded Radon measures. We characterize the center of the algebras $$\mathcal {M}(K)$$ , L p (K) and Trig(K) respectively, and consequently we obtain, for a certain class of hypergroups, the correspondence between the structure space of the center of L 1(K) and the center of Trig(K). As an application we study the existence of a finite universal Korovkin set w.r.t. positive operators in the center of L 1(K), in particular in L 1(K), whenever K is commutative.

Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier-Stokes equations

This article presents a family of very high-order non-uniform grid compact finite difference schemes with spatial orders of accuracy ranging from 4th to 20th for the incompressible Navier-Stokes eq...

A Compact Higher Order Finite Difference Method for the Incompressible Navier–Stokes Equations

A numerical method based on compact fourth order finite difference approximations is used for the solution of the incompressible Navier–Stokes equations. Our method is implemented for two dimensional, curvilinear coordinates on orthogonal, staggered grids. Two numerical experiments confirm the theoretically expected order of accuracy.