A Schur Complement Method for Compressible Navier-Stokes Equations

A posteriori finite element error estimation for second-order hyperbolic problems

Parallel computing of the transient radiative transfer process in the three-dimensional homogeneous and nonhomogeneous participating media is studied with an integral equation model. The model can be used for analyzing the ultra-short light pulse propagation within the highly scattering media. Two numerical quadratures are used: the discrete rectangular volume (DRV) method and YIX method. The parallel versions of both methods are developed for one-dimensional and three-dimensional geometries, respectively. Both quadratures achieve good speedup in parallel performance. Because the integral equation model uses very small amount of memory, the parallel computing can take advantage of having each compute node or processor store the full spatial domain information without using the typical domain decomposition parallelism, which will be necessary in other solution methods, e.g., discrete ordinates and finite volume methods, for large scale simulations. The parallel computation is conducted by assigning different portion of the quadrature to different compute node. In DRV method, a variation of the spatial domain decomposition is used. In the case of YIX scheme, the angular quadrature is divided up according to the number of compute nodes, instead of the spatial domain being divided. This parallel scheme minimizes the communications overhead. The only communication needed is at the end of each time step when each node shares the partial integrated result of the current time step with all other compute nodes. The angular quadrature decomposition approach leads to very good parallel efficiency. Two new discrete ordinate sets are used in the YIX angular quadrature and their parallel performances are discussed. One of the discrete ordinates sets, called spherical ring set, is also suitable for use in the conventional discrete ordinates method.

In the present work, flow characteristics of blood flow in the atherosclerotic right coronary artery in the presence of bi-directional magnetic field are investigated with numerical methods. The right coronary artery is one of two blood vessels that branch from the aorta and carry oxygen-rich blood to the heart muscle. Blockage of any branch of the coronary arteries causes death of a portion of the heart. A mathematical modelling is applied to analyse the pulsatile blood flow. Transient Navier-Stokes equations in 2D idealised arterial models of a bending artery coupled with Maxwell's equations for obtaining effect of magnetic field are discretised using the finite-volume method and solved by SIMPLE algorithm in curvilinear coordinate. The magnetic field is applied in both x and y directions. The amount of wall shear stress on the surface of atherosclerotic plaque is obtained for various times, and different states of magnetic field. Therefore, it is observed that how magnetic field modifies the flow patterns and decreases the destructive effects of atherosclerotic plaque.

We introduced in (Fortin & Serghini Mounim, 2005) a new method which allows us to extend the connection between the finite volume and dual mixed hybrid (DMH) methods to advection-diffusion problems in the one-dimensional case. In the present work we propose to extend the results of (Fortin & Serghini Mounim, 2005) to multidimensional hyperbolic and parabolic problems. The numerical approximation is achieved using the Raviart-Thomas (Raviart & Thomas, 1977) finite elements of lowest degree on triangular or rectangular partitions. We show the link with numerous finite volume schemes by use of appropriate numerical integrations. This will permit a better understanding of these finite volume schemes and the large number of DMH results available could carry out their analysis in a unified fashion. Furthermore, a stabilized method is proposed. We end with some discussion on possible extensions of our schemes.

The discontinuous Galerkin (dG) method was introduced by Reed and Hill [73] in 1973 for steady-state neutron transport as an hyperbolic problem. This was followed by other studies; by Bassi and Rebay [12] for the compressible Navier-Stokes equations, Cockburn and Shu [31] developed the local discontinuous Galerkin (ldG) method for advection-diffusion equations, and Peraire and Persson [69] introduced the compact discontinuous Galerkin (cdG) method. Independent of the dG methods, interior penalty (IP) methods have been developed for elliptic and parabolic problems by Douglas and Dupont [40] and Wheeler [95]. Then, in the 1980’s, Arnold et al. [6] proposed a unified classification and analysis of various kinds of dG methods. Later on, the dG methods were developed for elliptic problems [8, 24, 77] and for problems with advection [7, 14, 51, 56].

In recent years, there has been gratifying progress in the development of domain decomposition algorithms for symmetric and nonsymmetric elliptic problems and even some indefinite problems. Many methods possess the attractive property that the convergence rate is optimal, i.e., independent of the size of the discrete problem and of the number of subdomains, or within a polylog factor of optimal. There is, in comparison, relatively little in the domain decomposition literature on hyperbolic problems. Quarteroni [8, 9] used nonoverlapping domain decomposition methods based on the spectral collocation approximation on systems of conservation laws. Gastaldi and Gastaldi [5, 6] set up a nonoverlapping domain decomposition scheme based on the finite element approximation for the transport equation. These contributions establish the boundary operators that lead to well-posed decoupled problems, which can then be discretized and solved by standard means. Our interests in this paper are rather different. We examine overlapping domain decomposition preconditioners, and leave the original global discretization fully in tact. Rather than deriving interface conditions that lead to decomposed solutions that are mathematically equivalent (to within some specified discretization tolerance) to the solutions of the undecomposed problem, we derive an approximate inverse that can be applied in a concurrent manner, subdomain-by-subdomain, and that effectively preconditions the original undecomposed operator, whose action is already trivial to apply in the same concurrent manner. There seem to have been to date no such additive or multiplicative Schwarz preconditioners leading to optimal convergence rates for hyperbolic equations. Based on the standard Galerkin method [4] an ASM algorithm is formulated. The preconditioned problems are solved by the GMRES method. The convergence

Galerkin methods for a singularly perturbed hyperbolic problem with nonlocal nonlinearity

Computer Solution of Large Linear Systems

A priori error estimates are derived for several Galerkin procedures. For a nonlinear parabolic initial-boundary value problem, optimal rate L2-error estimates are derived for the forward and central difference Laplace modified Galerkin methods and the forward and central difference Laplace modified alternating direction Galerkin methods. For a nonlinear second order hyperbolic initial-boundary value problem, optimal rate L2-error estimates are derived for the Laplace modified Galerkin method and the Laplace modified alternating direction Galerkin method. 1. Introduction. In this paper we derive a priori error estimates for several Galerkin procedures. In ?3 we consider a nonlinear parabolic initial-boundary value problem, and we derive error estimates for the forward and central difference Laplace modified Galerkin methods and the forward and central difference Laplace modified alternating direction Galerkin methods of (5); these methods were formulated in (5), but in the central difference methods proofs were given only for the case in which the elliptic part was linear. In ?4 we consider a nonlinear second order hyperbolic initial-boundary value problem, and we derive error estimates for the Laplace modified Galerkin method and the Laplace modified alternating direction Galerkin method of (5); these methods are the obvious extensions of the methods of (5) to the nonlinear case. In ?2 we present certain notation, assumptions, and known results which are common to the development of ? 3 and 4. Section 5 contains some computational results.

In this work, we consider hyperbolic and parabolic evaluation problems on the Feynman-Dyson Hilbert space, $\mathcal{FD}_\otimes2$. We use some important properties given in $\mathcal{FD}_\otimes2$ to find solutions for both homogeneous and non-homogeneous cases. Therefore, we first focus the structure of the Feynman-Dyson Hilbert space from a mathematical perspective in terms of the construction of this space and the lifting of operator theory to this time-ordered setting. We then observe that $\mathcal{FD}_\otimes^2$ allows operators acting at different times to commute, while maintain their relative position on paper. We also deal with a time-ordered version of the Hille-Yosida theorem for semigroups of operators. This approach has the added advantage of requiring the weakest known domain and continuity conditions. We show these advantages for the generic classes of time-dependent homogeneous hyperbolic and parabolic problems. We also see that the theory has advantages for operators with no time dependence.

Multiscale finite-volume method for parabolic problems arising from compressible multiphase flow in porous media

are connected with the boundary conditions which do not fit into the classical theory of parabolic problems. The theory of parabolic (as well as elliptic and parameter-elliptic) problems even for scalar operators is deeply connected with the theory of mixed order system of pseudodierential operators acting on the boundary. The matrix-symbol of this system is the so-called Lopatinskii matrix. In standard parabolic problems, the corresponding system is parabolic in the sense of Solonnikov [8]. We introduce a more general class of parabolic boundary value problems replacing systems parabolic in the sense of [8] by a more general class of N-parabolic systems studied by the second author in [10]. In the definition of these systems the notion of Newton’s polygon plays a crucial role. It will be shown that the two mentioned examples from mathematical physics belong to this new class of boundary problems. We now come to the formulation of the two problems under consideration. For simplicity of presentation, we restrict ourselves to the model case of the half-space R n := {x = (x 0 ,xn) 2 R n : xn > 0} with boundary R n 1 . First, the linearized Stefan problem with Gibbs-Thompson correction is given by @tu(x,t) u(x,t) = f(x,t) (t > 0,x 2 R n ),

High-order, finite-volume methods in mapped coordinates

This paper describes the Domain Decomposition Method (DDM) for high frequency electromagnetic field problems. As an efficient iterative method to solve a complex symmetric system arising from the formulation of the E method, a DDM based on a MINimal RESidual (MINRES)-like iterative procedure is presented. Numerical experiments are demonstrated to report a faster convergence compared to conventional methods.

We study the iterated quasi-reversibility method to regularize ill-posed elliptic and parabolic problems: data completion problems for Poisson's and heat equations. We define an abstract setting to treat both equations at once. We demonstrate the convergence of the regularized solution to the exact one, and propose a strategy to deal with noise on the data. We present numerical experiments for both problems: a two-dimensional corrosion detection problem and the one-dimensional heat equation with lateral data. In both cases, the method prove to be efficient even with highly corrupted data.