Numerical Algorithms

In this paper, we study the spectral distributions of the saddle point matrices arising from the discretization and linearization of the Navier–Stokes equations, where the (1,1) block is nonsymmetric positive definite. In this paper, we derive the lower and upper bounds of the real and imaginary parts of all the eigenvalues of the saddle point matrices. We then propose a new class of block triangle preconditioners for solving the saddle point problems, and analyze the spectral properties of the preconditioned systems. Some numerical experiments with the preconditioned restarted generalized minimal residual method are reported to demonstrate the effectiveness and feasibility of these block triangle preconditioners.

One of the best known mechanisms of onset of chaotic motion is breaking of heteroclinic and homoclinic connections. It is well known that numerical integration on long time intervals very often becomes unstable (numerical instabilities) and gives rise to what is called “numerical chaos”. As one of the initial steps to discuss this phenomenon, we show in this paper that Euler's finite difference scheme does not preserve homoclinic connections.

We discuss convergence properties of the GMRES and FOM methods for solving large Sylvester equations of the form AX−XB=C. In particular we show the importance of the separation between the fields of values of A and B on the convergence behavior of GMRES. We also discuss the stagnation phenomenon in GMRES and its consequence on FOM. We generalize the issue of breakdown in the block-Arnoldi algorithm and explain its consequence on FOM and GMRES methods. Several numerical tests illustrate the theoretical results.

In this paper, we define s-nonstationary iterative process and obtain its properties. We prove, that for any one-point iterative process without memory, there exists an s-nonstationary process of the same order, but of higher efficiency by the criteria of Traub and Ostrowski. We supply constructions of s-nonstationary processes for Newton’s, Halley’s, and Chebyshev’s methods, obtain their properties and, for some of them, also their geometric interpretation. The algorithms we present can be transformed into computer programs in a straightforward manner. Additionally, we illustrate numerical examples, as demonstrations for the methods we present.

In this paper, we consider the generalized Kronecker product (GKP) linear system associated with a class of consecutive-rank-descending (CRD) matrices arising from bivariate interpolation problems. Relying on the sign sequences of CRD matrices, we show that the associated GKP linear system is accurately solved with an “ideal” componentwise forward error. In particular, a pleasantly small componentwise relative forward error is provided to illustrate that each component of the solution is computed to high relative accuracy. We then present the sign sequences of generalized Vandermonde matrices to show that the associated GKP linear system is accurately solved with the desired componentwise forward errors. Numerical experiments are performed to confirm the high relative accuracy.

This paper is devoted to condition numbers of the total least squares problem with linear equality constraint (TLSE). With novel limit techniques, closed formulae for normwise, mixed and componentwise condition numbers of the TLSE problem are derived. Computable expressions and upper bounds for these condition numbers are also given to avoid the costly Kronecker product-based operations. The results unify the ones for the TLS problem. For TLSE problems with equilibratory input data, numerical experiments illustrate that normwise condition number-based estimate is sharp to evaluate the forward error of the solution, while for sparse and badly scaled matrices, mixed and componentwise condition number-based estimates are much tighter.

In this article, we mainly consider a first-order decoupling penalty method for the 2D/3D time-dependent incompressible magnetohydrodynamic (MHD) equations in a convex domain. This method applies a penalty term to the constraint “divu = 0,” which allows us to transform the saddle point problem into two small problems to solve. The time discretization is based on the backward Euler scheme. Moreover, we derive the optimal error estimate for the penalty method under semi-discretization with the relationship 𝜖 = O(Δt). Finally, we give abundant of numerical tests to verify the theoretical result and the spatial discretization is based on Lagrange finite element.