Solution of nonlinear convection-diffusion problems by a conservative Galerkin-characteristics method

We consider a scheme for nonlinear (degenerate) convection dominant diffusion problems that arise in contaminant transport in porous media with equilibrium adsorption isotherm. This scheme is based on a regularization relaxation scheme that has been introduced by Jäger and Ka?ur (Numer Math 60:407–427, 1991; M²AN Math Model Numer Anal 29(N5):605–627, 1995) with a type of numerical integration by Bermejo (SIAM J Numer Anal 32:425–455, 1995) to the modified method of characteristics with adjusted advection MMOCAA that was recently developed by Douglas et al. (Numer Math 83(3):353–369, 1999; Comput Geosci 1:155–190, 1997). We present another variant of adjusting advection method. The convergence of the scheme is proved. An error estimate of the approximated scheme is derived. Computational experiments are carried out to illustrate the capability of the scheme to conserve the mass.     

Convergence of finite volume schemes for triangular systems of conservation laws

We consider non-strictly hyperbolic systems of conservation laws in triangular form, which arise in applications like three-phase flows in porous media. We device simple and efficient finite volume schemes of Godunov type for these systems that exploit the triangular structure. We prove that the finite volume schemes converge to weak solutions as the discretization parameters tend to zero. Some numerical examples are presented, one of which is related to flows in porous media. The research of K. H. Karlsen was supported by an Outstanding Young Investigators Award from the Research Council of Norway.     

Robust convergence of a compact fourth-order finite difference scheme for reaction–diffusion problems

We consider a singularly perturbed one-dimensional reaction–diffusion problem with strong layers. The problem is discretized using a compact fourth order finite difference scheme. Altough the discretization is not inverse monotone we are able to establish its maximum-norm stability and to prove its pointwise convergence on a Shishkin mesh. The convergence is uniform with respect to the perturbation parameter. Numerical experiments complement our theoretical results.     

Fundamental bounds for steady water waves

The two-dimensional free-boundary problem of steady gravity waves on water of finite depth is considered. Bounds on the free-surface profiles and on the values of Bernoulli’s constant are obtained under minimal assumptions about properties of solutions to the problem.     

A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models

The paper is concerned with development of a new finite-volume method for a class of chemotaxis models and for a closely related haptotaxis model. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. The first step in the derivation of the new method is made by adding an equation for the chemoattractant concentration gradient to the original system. We then show that the convective part of the resulting system is typically of a mixed hyperbolic-elliptic type and therefore straightforward numerical methods for the studied system may be unstable. The proposed method is based on the application of the second-order central-upwind scheme, originally developed for hyperbolic systems of conservation laws in Kurganov et al. (SIAM J Sci Comput 21:707–740, 2001), to the extended system of PDEs. We show that the proposed second-order scheme is positivity preserving, which is a very important stability property of the method. The scheme is applied to a number of two-dimensional problems including the most commonly used Keller–Segel chemotaxis model and its modern extensions as well as to a haptotaxis system modeling tumor invasion into surrounding healthy tissue. Our numerical results demonstrate high accuracy, stability, and robustness of the proposed scheme.     

Convergence analysis of the high-order mimetic finite difference method

We prove second-order convergence of the conservative variable and its flux in the high-order MFD method. The convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients. For the case of non-constant coefficients, we also develop a new family of high-order MFD methods. Theoretical result are confirmed through numerical experiments.     

Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods

We derive in this paper a posteriori error estimates for discretizations of convection–diffusion–reaction equations in two or three space dimensions. Our estimates are valid for any cell-centered finite volume scheme, and, in a larger sense, for any locally conservative method such as the mimetic finite difference, covolume, and other. We consider meshes consisting of simplices or rectangular parallelepipeds and also provide extensions to nonconvex cells and nonmatching interfaces. We allow for the cases of inhomogeneous and anisotropic diffusion–dispersion tensors and of convection dominance. The estimates are established in the energy (semi)norm for a locally postprocessed approximate solution preserving the conservative fluxes and are of residual type. They are fully computable (all occurring constants are evaluated explicitly) and locally efficient (give a local lower bound on the error times an efficiency constant), so that they can serve both as indicators for adaptive refinement and for the actual control of the error. They are semi-robust in the sense that the local efficiency constant only depends on local variations in the coefficients and becomes optimal as the local Péclet number gets sufficiently small. Numerical experiments confirm their accuracy. This work was supported by the GdR MoMaS project “Numerical Simulations and Mathematical Modeling of Underground Nuclear Waste Disposal”, PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN, France. The main part of this work was carried out during the author’s post-doc stay at Laboratoire de Mathématiques, Analyse Numérique et EDP, Université de Paris-Sud and CNRS, Orsay, France.     

Error estimates of proper orthogonal decomposition eigenvectors and Galerkin projection for a general dynamical system arising in fluid models

We study estimates for proper orthogonal decomposition eigenvectors and eigenvalues as well as error estimates between the exact solution of a 2D Navier–Stokes model and the numerical approach when the proper orthogonal decomposition method is considered. These estimates are also extended when bifurcation diagram are calculated using the so called p-POD or SPOD methods with a new cut-off criterion to minimize noisy modes produced by the p-POD method.     

Convergence of lattice Boltzmann methods for Navier–Stokes flows in periodic and bounded domains

Combining an asymptotic analysis of the lattice Boltzmann method with a stability estimate, we are able to prove some convergence results which establish a strict relation to the incompressible Navier–Stokes equation. The proof applies to the lattice Boltzmann method in the case of periodic domains and for specific bounded domains if the Dirichlet boundary condition is realized with the bounce back rule.     

Existence of weak solution and semiclassical limit for quantum drift-diffusion model

The existence and semiclassical limit of the solution to one-dimensional transient quantum drift-diffusion model in semiconductor simulation are discussed. Besides the proof of existence of the weak solution, it is also obtained that the semiclassical limit of this solution solves the classical drift-diffusion model. The key estimates rest on the entropy inequalities derived from separation of quantum quasi-Fermi level.     

The rate of convergence of GMRES on a tridiagonal Toeplitz linear system

The Generalized Minimal Residual method (GMRES) is often used to solve a nonsymmetric linear system Ax = b. But its convergence analysis is a rather difficult task in general. A commonly used approach is to diagonalize A = XΛ X ⁻¹ and then separate the study of GMRES convergence behavior into optimizing the condition number of X and a polynomial minimization problem over A’s spectrum. This artificial separation could greatly overestimate GMRES residuals and likely yields error bounds that are too far from the actual ones. On the other hand, considering the effects of both A’s spectrum and the conditioning of X at the same time poses a difficult challenge, perhaps impossible to deal with in general but only possible for certain particular linear systems. This paper will do so for a (nonsymmetric) tridiagonal Toeplitz system. Sharp error bounds on and sometimes exact expressions for residuals are obtained. These expressions and/or bounds are in terms of the three parameters that define A and Chebyshev polynomials of the first kind.     

Scattering of an incident acoustic wave by an elastic sphere in a shallow water

The scattering of acoustic waves by an elastic sphere in a shallow ocean wave guide is investigated taking into account the shear waves which can exist in addition to compressional waves in scatterers of solid material. Expressions for the scattered waves are given. Numerical values for a quantity called the farfield form function for various depth are presented in graphical forms.      

Conservation of energy,momentum and actions in numerical discretizations of non-linear wave equations

For classes of symplectic and symmetric time-stepping methods— trigonometric integrators and the Störmer–Verlet or leapfrog method—applied to spectral semi-discretizations of semilinear wave equations in a weakly non-linear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time.     

A hybrid method for computing Lyapunov exponents

In this paper we propose a numerical method for computing all Lyapunov coefficients of a discrete time dynamical system by spatial integration. The method extends an approach of Aston and Dellnitz (Comput Methods Appl Mech Eng 170:223–237, 1999) who use a box approximation of an underlying ergodic measure and compute the first Lyapunov exponent from a spatial average of the norms of the Jacobian for the iterated map. In the hybrid method proposed here, we combine this approach with classical QR-oriented methods by integrating suitable R-factors with respect to the invariant measure. In this way we obtain approximate values for all Lyapunov exponents. Assuming somewhat stronger conditions than those of Oseledec’ multiplicative theorem, these values satisfy an error expansion that allows to accelerate convergence through extrapolation. W.-J. Beyn and A. Lust was supported by CRC 701 ‘Spectral Analysis and Topological Methods in Mathematics’. The paper is mainly based on the PhD thesis [27] of A. Lust.     

Mathematical and numerical study of a system of conservation laws

The system of equations (f (u))_t − (a(u)v + b(u))_x = 0 and u_t − (c(u)v + d(u))_x = 0, where the unknowns u and v are functions depending on , arises within the study of some physical model of the flow of miscible fluids in a porous medium. We give a definition for a weak entropy solution (u, v), inspired by the Liu condition for admissible shocks and by Krushkov entropy pairs. We then prove, in the case of a natural generalization of the Riemann problem, the existence of a weak entropy solution only depending on x/t. This property results from the proof of the existence, by passing to the limit on some approximations, of a function g such that u is the classical entropy solution of u_t − ((cg + d)(u))_x = 0 and simultaneously w = f (u) is the entropy solution of w_t − ((ag + b)(f⁽⁻¹⁾(w)))_x = 0. We then take v = g(u), and the proof that (u, v) is a weak entropy solution of the coupled problem follows from a linear combination of the weak entropy inequalities satisfied by u and f (u). We then show the existence of an entropy weak solution for a general class of data, thanks to the convergence proof of a coupled finite volume scheme. The principle of this scheme is to compute the Godunov numerical flux with some interface functions ensuring the symmetry of the finite volume scheme with respect to both conservation equations.     

On the first eigenvalue of a fourth order Steklov problem

We prove some results about the first Steklov eigenvalue d ₁ of the biharmonic operator in bounded domains. Firstly, we show that Fichera’s principle of duality (Fichera in Atti Accad Naz Lincei 19:411–418, 1955) may be extended to a wide class of nonsmooth domains. Next, we study the optimization of d ₁ for varying domains: we disprove a long-standing conjecture, we show some new and unexpected features and we suggest some challenging problems. Finally, we prove several properties of the ball.     

Convergence of LR algorithm for a one-point spectrum tridiagonal matrix

We prove convergence for the basic LR algorithm on a real unreduced tridiagonal matrix with a one-point spectrum—the Jordan form is one big Jordan block. First we develop properties of eigenvector matrices. We also show how to deal with the singular case.     

The Nielsen number for free fundamental groups and maps without remnant

For Y any space that has the homotopy type of a wedge of finitely many circles, and for g : Y → Y a map, the Nielsen number of g, N(g), is a homotopy invariant lower bound for the size of the fixed point set of any map homotopic to g. Such a map g has k-remnant if, roughly, there is limited cancellation in any product g _♯(u)g _♯(v) where g _♯ is the induced homomorphism and u, v ∈ π₁(Y) and |u| = |v| = k. We prove that such maps are (k + 1)-characteristic, meaning that in order to determine the Nielsen classes of fixed points, we need only test whether a limited, specified, set of elements z ∈ π₁(Y) are solutions to the equation z = W ⁻¹ _x f _♯(z)W _y , with x and y fixed points that are represented in the fundamental group by W _x and W _y , respectively. The number of elements to be tested is profoundly decreased by using abelianization as well. This work is a significant extension of Wagner’s results involving maps with remnant and Wagner’s algorithm. Our proofs involve new concepts and techniques. We present an algorithm for N(g) for any map g with k-remnant, and we provide examples for which no algebraic techniques previously known would work. One example shows that for any k there is a map that does not have (k − 1)-remnant but does have k-remnant. Dedicated to Edward Fadell for inspirational teaching and guidance as the thesis advisor of the first author     

Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

The integrability of an m-component system of hydrodynamic type, u _t = V(u)u _x , by the generalized hodograph method requires the diagonalizability of the m ×  m matrix V(u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains—infinite-component systems of hydrodynamic type for which the ∞ ×  ∞ matrix V(u) is ‘sufficiently sparse’. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.     

On approximate solutions to the first initial boundary value problem for the m-Hessian evolution equations

We adapt to degenerate m-Hessian evolution equations the notion of m-approximate solutions introduced by N. Trudinger for m-Hessian elliptic equations, and we present close to necessary and sufficient conditions guaranteeing the existence and uniqueness of such solutions for the first initial boundary value problem. Dedicated to Professor Felix Browder