A regularization-stabilization technique for nonlinear conservation equation computations

A regularization procedure for nonlinear conservation equations is introduced and demonstrated to have a stabilizing effect on the numerical solution of the associated approximate problem. Representative results for a least-squares finite-element method are given, and the numerical performance of the stabilization procedure explored. The effect of the regularization term is similar to a local numerical dissipation dependent on the numerical itegration time step.     

A mapping technique for numerical computations of bed evolutions

Free surface flow is one of the most difficult problems in engineering to be solved, since velocity and pressure fields depend on the free surface. On the other hand, the position of the free surface is unknown previously. Furthermore, the boundary condition on the free surface is expressed by a complicated equation. In an alluvial stream, where the boundaries of the domain are not fixed, addition of free surface at the bed will increase this difficulty. A domain mapping technique is developed in this paper to study the bed evolutions. The flow is considered 2D, choosing two coordinates in streamwise and upward directions. With a proper transformation, the hydrodynamics and sediment transport governing equations in irregular domain will be mapped into a simple rectangular one. The new domain can be discretize by finite elements. The transformed governing equations are solved to obtain desired variables in the mapped domain. With a proper transformation, there is no need of inverse mapping to obtain the free water surface profile and bedform evolution and migration in the actual domain. The model has been applied to streams with movable bed and the results show a good agreement with the experimental experiences.     

A strict Positivstellensatz for enveloping algebras

Let G be a connected and simply connected real Lie group with Lie algebra . Semialgebraic subsets of the unitary dual of G are defined and a strict Positivstellensatz for positive elements of the universal enveloping algebra of is proved. An erratum to this article is available at .     

The combination technique for a two-dimensional convection-diffusion problem with exponential layers

Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing N for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on and meshes. It is shown that the combination FEM yields (up to a factor ln N) the same order of accuracy in the associated energy norm as the Galerkin FEM on an N × N mesh, but it requires only (N ^3/2) degrees of freedom compared with the (N ²) used by the Galerkin FEM. An analogous result is also proved for the streamline diffusion finite element method. This work was supported by the National Natural Science Foundation of China (10701083 and 10425105), the Chinese National Basic Research Program (2005CB321704) and the Boole Centre for Research in Informatics at National University of Ireland Cork.     

A posteriori error estimators for convection-diffusion equations

Summary. We derive a posteriori error estimators for convection-diffusion equations with dominant convection. The estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh-Peclet number. The estimators are either based on the evaluation of local residuals or on the solution of discrete local Dirichlet or Neumann problems. Received February 10, 1997 / Revised version received November 4, 1997     

Application of regularization technique to variational adjoint method: A case for nonlinear convection-diffusion problem

A discrete assimilation system for a one-dimensional variable coefficient convection-diffusion equation is constructed. The variational adjoint method combined with the regularization technique is employed to retrieve the initial condition and diffusion coefficient with the aid of a set of simulated observations. Several numerical experiments are performed: (a) retrieving both the initial condition and diffusion coefficient jointly (Experiment JR), (b) retrieving either of them separately (Experiment SR), (c) retrieving only the diffusion coefficient with the iteration count increased to 800 (Experiment NoR-SR), and (d) retrieving only the diffusion coefficient with the consideration of a regularization term based on the Experiment NoR-SR (Experiment AdR-SR). The results indicate that within the limit of 100 iterations, the retrieval quality of the Experiment SR is better than those from the Experiment JR. Compared with the initial condition, the diffusion coefficient is a little difficult to retrieve, whereas we still achieve the desired result by increasing the iterations or integrating the regularization term into the cost functional for the improvement with respect to the diffusion coefficient. Further comparisons between the Experiment NoR-SR and AdR-SR show that the regularization term can really help not only improve the precision of retrieval to a large extent, but also speed up the convergence of solution, even if some perturbations are imposed on those observations.     

A semi-circulant preconditioner for the convection-diffusion equation

In this paper we define and analyze a semi-circulant preconditioner for the convection-diffusion equation. We derive analytical formulas for the eigenvalues and the eigenvectors of the preconditioned system of equations. We show that for mesh Péclet numbers less than 2, the rate of convergence depends only on the mesh Péclet number and the direction of the convective field and not on the spatial grid ratio or the number of unknowns. Received February 20, 1997 / Revised version received November 19, 1997     

A nonlocal convection-diffusion equation

In this paper we study a nonlocal equation that takes into account convective and diffusive effects, ut=J∗u−u+G∗(f(u))−f(u) in Rd, with J radially symmetric and G not necessarily symmetric. First, we prove existence, uniqueness and continuous dependence with respect to the initial condition of solutions. This problem is the nonlocal analogous to the usual local convection-diffusion equation ut=Δu+b⋅∇(f(u)). In fact, we prove that solutions of the nonlocal equation converge to the solution of the usual convection-diffusion equation when we rescale the convolution kernels J and G appropriately. Finally we study the asymptotic behaviour of solutions as t→∞ when f(u)=|u|^q−1u with q>1. We find the decay rate and the first-order term in the asymptotic regime.     

Sections of a differential spectrum and factorization-free computations

We construct sections of a differential spectrum using only localization and projective limits. For this purpose we introduce a special form of multiplicative systems generated by one differential polynomial and call it D-localization. Owing to this technique one can construct sections of a differential spectrum of a differential ring without computation of diffspec . We compare our construction with Kovacic’s structure sheaf and with the results obtained by Keigher [J. Pure Appl. Algebra, 27, 163–172 (1983)]. We show how to compute sections of factor-rings of rings of differential polynomials. All computations in this paper are factorization-free. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 133–144, 2003.     

The prime spectrum of the enveloping algebra of a reductive Lie algebra

Written during the author's stay at MSRI, supported by a Stipendium der Clemens Plassmann Stiftung     

对流扩散方程的一种新型特征差分方法

利用在网格内恰当选取特征线上插值点的技巧,提出了一种新型的求解对流扩散方程的特征差分方法,并给出了稳定性与收敛性分析.该方法避免了数值扩散的产生,同时具有O(τ h~2)阶的收敛阶.数值实验表明,该方法是一个高效、稳定和收敛的数值方法.     

A robust discontinuous Galerkin method for solving convection-diffusion problems

In this paper, a new DG method was designed to solve the model problem of the one-dimensional singularly-perturbed convection-diffusion equation. With some special chosen numerical traces, the existence and uniqueness of the DG solution is provided. The superconvergent points inside each element are observed. Particularly, the 2p ＋ 1-order superconvergence and even uniform superconvergence under layer-adapted mesh are observed numerically.     

Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems

Summary. We analyze nonconforming finite element approximations of streamline-diffusion type for solving convection-diffusion problems. Both the theoretical and numerical investigations show that additional jump terms have to be added in the nonconforming case in order to get the same order of convergence in L as in the conforming case for convection dominated problems. A rigorous error analysis supported by numerical experiments is given. Received June 26, 1996 / Revised version received November 20, 1996     

A second order characteristic finite element scheme for convection-diffusion problems

Summary. A new characteristic finite element scheme is presented for It is of second order accuracy in time increment, symmetric, and unconditionally stable. Optimal error estimates are proved in the framework of -theory. Numerical results are presented for two examples, which show the advantage of the scheme. Received November 22, 2000 / Revised version received July 11, 2001 / Published online October 17, 2001     

A characteristic difference method for the transient fractional convection-diffusion equations

A new characteristic finite difference method for solving the two-sided space-fractional convection-diffusion equations is presented, by combining characteristic methods and fractional finite difference methods. Stability, consistency and (therefore) convergence of the new method are discussed in this paper. An error estimate is given. Numerical experiments of this method are carried out and compared with other known methods.     

A characteristic-mixed finite element method for time-dependent convection-diffusion optimal control problem

In this paper, we examine the method of characteristic-mixed finite element for the approximation of convex optimal control problem governed by time-dependent convection-diffusion equations with control constraints. For the discretization of the state equation, the characteristic finite element is used for the approximation of the material derivative term (i.e., the time derivative term plus the convection term), and the lowest-order Raviart-Thomas mixed element is applied for the approximation of the diffusion term. We derive some a priori error estimates for both the state and control approximations.     

A fractional step θ-method for convection-diffusion problems

In this article, we analyze the fractional step θ-method for the time-dependent convection-diffusion equation. In our implementation, we completely separate the convection operator from the diffusion operator, and stabilize the convective problem using a Streamline Upwinded Petrov-Galerkin (SUPG) method. We establish a priori error estimates and show that the optimal value of θ yields a scheme that is second-order in time. Numerical computations are presented which demonstrate the method and support the theoretical results.     

A note on Hermitian splitting induced relaxation methods for convection-diffusion equations

The solution of the linear system Ax = b by iterative methods requires a splitting of the coefficient matrix in the form A = M − N where M is usually chosen to be a diagonal or a triangular matrix. In this article we study relaxation methods induced by the Hermitian and skew-Hermitian splittings for the solution of the linear system arising from a compact fourth order approximation to the one dimensional convection-diffusion equation and compare the convergence rates of these relaxation methods to that of the widely used successive overrelaxation (SOR) method. Optimal convergence parameters are derived for each method and numerical experiments are given to supplement the theoretical estimates. For certain values of the diffusion parameter, a relaxation method based on the Hermitian splitting converges faster than SOR. For two-dimensional problems a block form of the iterative algorithm is presented. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 581–591, 1998