A least-squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative-norm least-squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax–Wendroff theorem. The numerical conservation properties of the method in this paper do not fall precisely in the framework introduced by Lax and Wendroff, but they are similar in spirit as they guarantee that when L² convergence holds, the resulting approximations approach a weak solution to the hyperbolic problem. The least-squares functional is continuous and coercive in an H⁻¹-type norm, but not L²-coercive. Nevertheless, the L² convergence properties of the method are discussed. Convergence can be obtained either by an explicit regularization of the functional, that provides control of the L² norm, or by properly choosing the finite element spaces, providing implicit control of the L² norm. Numerical results for the inviscid Burgers equation with discontinuous source terms are shown, demonstrating the L² convergence of the obtained approximations to the physically admissible solution. The numerical method utilizes a least-squares functional, minimized on finite element spaces, and a Gauss–Newton technique with nested iteration. We believe that the linear systems encountered with this formulation are amenable to multigrid techniques and combining the method with adaptive mesh refinement would make this approach an efficient tool for solving balance laws (this is the focus of a future study).     

Global weak solution of planetary geostrophic equations with inviscid geostrophic balance

A reformulation of the planetary geostrophic equations (PGEs) with the inviscid balance equation is proposed and the existence of global weak solutions is established, provided that the mechanical force satisfies an integral constraint. There is only one prognostic equation for the temperature field, and the velocity field is statically determined by the planetary geostrophic balance combined with the incompressibility condition. Furthermore, the velocity profile can be accurately represented as a function of the temperature gradient. In particular, the vertical velocity depends only on the first-order derivative of the temperature. As a result, the bound for the L∞ (0, t ₁ ; L ²) ∩ L ² (0, t ₁ ; H ¹) norm of the temperature field is sufficient to show the existence of the weak solution.     

Numerical approximation of entropy solutions for hyperbolic integro-differential equations

Summary. Scalar hyperbolic integro-differential equations arise as models for e.g. radiating or self-gravitating fluid flow. We present finite volume schemes on unstructured grids applied to the Cauchy problem for such equations. For a rather general class of integral operators we show convergence of the approximate solutions to a possibly discontinuous entropy solution of the problem. For a specific model problem in radiative hydrodynamics we introduce a convergent fully discrete finite volume scheme. Under the assumption of sufficiently fast spatial decay of the entropy solution we can even establish the convergence rate h^1/4|ln(h)| where h denotes the grid parameter. The convergence proofs rely on appropriate variants of the classical Kruzhkov method for local balance laws together with a truncation technique to cope with the nonlocal character of the integral operator.Mathematics Subject Classification (2000): 35L65, 35Q35, 65M15     

The stationary Stefan problem with convection governed by a non‐linear Darcy's law

We consider the bidimensional stationary Stefan problem with convection. The problem is governed by a coupled system involving a non‐linear Darcy's law and the energy balance equation with second member in L¹. We prove existence of at least one weak solution of the problem, using the penalty method and the Schauder fixed point principle. Copyright © 1999 John Wiley & Sons, Ltd.     

On a backward problem for fractional diffusion equation with Riemann-Liouville derivative

In the present paper, we study the initial inverse problem (backward problem) for a two-dimensional fractional differential equation with Riemann-Liouville derivative. Our model is considered in the random noise of the given data. We show that our problem is not well-posed in the sense of Hadamard. A truncated method is used to construct an approximate function for the solution (called the regularized solution). Furthermore, the error estimate of the regularized solution in L² and H^τ norms is considered and illustrated by numerical example.     

Computable analysis of a boundary‐value problem for the generalized KdV–Burgers equation

In this paper, we investigate the computability of the solution operator of the generalized KdV‐Burgers equation with initial‐boundary value problem. Here, the solution operator is a nonlinear map H^{3m ? 1}(R⁺) × H^m(0,T)→C([0,T];H^{3m ? 1}(R⁺)) from the initial‐boundary value data to the solution of the equation. By a technique that is widely used for the study of nonlinear dispersive equation, and using the type 2 theory of effectivity as computable model, we prove that the solution map is Turing computable, for any integer m ≥ 2, and computable real number T > 0. Copyright © 2014 John Wiley & Sons, Ltd.     

ΓD‐convergence for a class of quasilinear elliptic equations in thin structures

We study the asymptotic behaviour of the solution of a stationary quasilinear elliptic problem posed in a domain Ω^(ε) of asymptotically degenerating measure, i.e. meas Ω^(ε) → 0 as ε → 0, where ε is the parameter that characterizes the scale of the microstructure. We obtain the convergence of the solution and the homogenized model of the problem is constructed using the notion of convergence in domains of degenerating measure. Proofs are given using the method of local characteristics of the medium Ω^(ε) associated with our problem in a variational form. Copyright © 2005 John Wiley & Sons, Ltd.     

The general common Hermitian nonnegative-definite solution to the matrix equations AXA=BB and CXC=DD with applications in statistics

We deduce a necessary and sufficient condition for the matrix equations AXA^*=BB^* and CXC^*=DD^* to have a common Hermitian nonnegative-definite solution and a representation of the general common Hermitian nonnegative-definite solution to these two equations when they have such common solutions. Thereby, we solve a statistical problem which is concerned in testing linear hypotheses about regression coefficients in the multivariate linear model. This paper is a revision of Young et al. (J. Multivariate Anal. 68 (1999) 165) whose mistake was pointed out in (Linear Algebra Appl. 321 (2000) 123).     

Machine interference with general repair and running times

Machine maintenance and repair operations, alternatively called machine interference, may be effectively represented as finite source queueing processes. This paper presents a general solution to such processes where both the repair times and running (operating) times are modelled as Erlang distributions (the (E _h/E_m/1)/K model). Such a general solution is unavailable elsewhere. The solution is made possible by two devices. Firstly the finite source process is replaced by an equivalent two stage cyclic queueing process where the two stages are the repair stage and the operating stage. Secondly a one-dimensional state definition is used for each stage giving a total two-dimensional state definition for the whole process. This two-dimensional state permits the straightforward visualization and drawing of the rate diagram and the consequent extraction of the steady state balance equations from this rate diagram. A numerical solution procedure for the balance equations is also outlined as is an extension to the multiple repair facility case (the (E _h/E_m/c)/K model).     

On the nonuniqueness of weak solution of the Euler equation

Weak solution of the Euler equations is defined as an L²-vector field satisfying the integral relations expressing the mass and momentum balance. Their general nature has been quite unclear. In this work an example of a weak solution on a 2-dimensional torus is constructed that is identically zero outside a finite time interval. This example is simpler and more transparent than the previous example of V. Scheffer (J. Geom. Anal. 3(4), 1993, pp. 343–401). © 1997 John Wiley & Sons, Inc.     

A New and Pragmatic Approach to the GIX/Geo/c/N Queues Using Roots

A simple and complete solution to determine the distributions of queue lengths at different observation epochs for the model GI^X/Geo/c/N is presented. In the past, various discrete-time queueing models, particularly the multi-server bulk-arrival queues with finite-buffer have been solved using complicated methods that lead to results in a non-explicit form. The purpose of this paper is to present a simple derivation for the model GI^X/Geo/c/N that leads to a complete solution in an explicit form. The same method can also be used to solve the GI^X/Geo/c/N queues with heavy-tailed inter-batch-arrival time distributions. The roots of the underlying characteristic equation form the basis for all distributions of queue lengths at different time epochs. All queue-length distributions are in the form of sums of geometric terms.

     

Asymptotics of the 1/2-dimension,Discrete Airy Equation and its Approximating Differential Equation

A discrete finite difference model is constructed for the Airy equation using a nonstandard scheme formulated by Mickens and Ramadhani. The method of dominant balance is then applied to obtain a first-order difference equation for the solution that increases sufficiently fast as k→∞. We then calculate the corresponding approximating differential equation and obtain its exact solution as well as its “exact” discrete finite difference representation. The application of various symmetry operations allows the determination of the related rapidly decreasing solution and the oscillatory solutions for negative values of x k>=hk, where h=?x.     

Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy

We study the asymptotic time behavior of global smooth solutions to general entropy, dissipative, hyperbolic systems of balance laws in m space dimensions, under the Shizuta‐Kawashima condition. We show that these solutions approach a constant equilibrium state in the L^p‐norm at a rate O(t^{? (m/2)(1 ? 1/p)}) as t → ∞ for p ∈ [min{m, 2}, ∞]. Moreover, we can show that we can approximate, with a faster order of convergence, the conservative part of the solution in terms of the linearized hyperbolic operator for m ≥ 2, and by a parabolic equation, in the spirit of Chapman‐Enskog expansion in every space dimension. The main tool is given by a detailed analysis of the Green function for the linearized problem. © 2007 Wiley Periodicals, Inc.     

Time decay of the solution to the Cauchy problem for a three-dimensional model of nonsimple thermoelasticity

This paper is devoted to the investigation of the solution to the Cauchy problem for a system of partial differential equations describing thermoelasticity of nonsimple materials in a three-dimensional space. The model of linear dynamical thermoelasticity of nonsimple materials is considered as the system of partial differential equations of fourth order. In this paper, we proposed a convenient evolutionary method of approach to the system of equations of nonsimple thermoelasticity. We proved the L^p−L^q time decay estimates for the solution to the Cauchy problem for linear thermoelasticity of nonsimple materials.     

Hyperbolic heat conduction in two semi-infinite bodies in contact

We find, under the viewpoint of the hyperbolic model of heat conduction, the exact analytical solution for the temperature distribution in all points of two semi-infinite homogeneous isotropic bodies that initially are at uniform temperatures T ₀ ¹ and T ₀ ² , respectively, suddenly placed together at time t = 0 and assuming that the contact between the bodies is perfect. We make graphics of the obtained temperature profiles of two bodies at different times and points. And finally, we compare the temperature solution obtained from hyperbolic model to the parabolic or classical solution, for the same problem of heat conduction.This work was partially supported by MEC and FEDER, project MTM-2004-02262 and AVCIT group 03/050.This revised version was published online in April 2005 with a corrected issue number.     

Stability and convergence for a complete model of mass diffusion

We propose a fully discrete scheme for approximating a three-dimensional, strongly nonlinear model of mass diffusion, also called the complete Kazhikhov–Smagulov model. The scheme uses a C⁰ finite-element approximation for all unknowns (density, velocity and pressure), even though the density limit, solution of the continuous problem, belongs to H². A first-order time discretization is used such that, at each time step, one only needs to solve two decoupled linear problems for the discrete density and the velocity–pressure, separately.We extend to the complete model, some stability and convergence results already obtained by the last two authors for a simplified model where λ²-terms are not considered, λ being the mass diffusion coefficient. Now, different arguments must be introduced, based mainly on an induction process with respect to the time step, obtaining at the same time the three main properties of the scheme: an approximate discrete maximum principle for the density, weak estimates for the velocity and strong ones for the density. Furthermore, the convergence towards a weak solution of the density-dependent Navier–Stokes problem is also obtained as λ→0 (jointly with the space and time parameters).Finally, some numerical computations prove the practical usefulness of the scheme.     

Mathematical and physical aspect of the solution of the Cauchy problem for a system of equations describing a nonlocal model of propagation of heat with finite speed

In our paper we present a new system of equations describing a nonlocal model of propagation of heat with finite speed in three-dimensional space. Such a system of equations is described by a system of integral – differential equations. At first using the modiffied Cagniard de Hoop method, we construct the fundamental solution of this system of equations. On the basis of the constructed fundamental solution we obtain the explicite formulate of the solution of the Cauchy problem for this system of equations and applying the method of Sobolev and Biesov spaces, we get L^p – L^q time decay estimate for the solution of the Cauchy problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Global Existence of Small Solutions to the Oldroyd-B Model

The Cauchy problem to the Oldroyd-B model is studied. In particular, it is shown that if the smooth solution (u, τ) to this system blows up at a finite time T*, then ∫₀ ^T* ‖▿u(t)‖ _L ^∞dt = ∞. Furthermore, the global existence of smooth solution to this system is given with small initial data.     

Invariant manifolds for weak solutions to stochastic equations

Viability and invariance problems related to a stochastic equation in a Hilbert space H are studied. Finite dimensional invariant C ² submanifolds of H are characterized. We derive Nagumo type conditions and prove a regularity result: any weak solution, which is viable in a finite dimensional C ² submanifold, is a strong solution. These results are related to finding finite dimensional realizations for stochastic equations. There has recently been increased interest in connection with a model for the stochastic evolution of forward rate curves. Received: 15 April 1999 / Revised version: 4 February 2000 / Published online: 18 September 2000