Fundamental mode exact schemes for unsteady problems

The problem of increasing the accuracy of an approximate solution is considered for boundary value problems for parabolic equations. For ordinary differential equations (ODEs), nonstandard finite difference schemes are in common use for this problem. They are based on a modification of standard discretizations of time derivatives and, in some cases, allow to obtain the exact solution of problems. For multidimensional problems, we can consider the problem of increasing the accuracy only for the most important components of the approximate solution. In the present work, new unconditionally stable schemes for parabolic problems are constructed, which are exact for the fundamental mode. Such two‐level schemes are designed via a modification of standard schemes with weights using Padé approximations. Numerical results obtained for a model problem demonstrate advantages of the proposed fundamental mode exact schemes.     

Efficient solutions schemes for interface problems

A computationally efficient two-level iterative scheme is proposed for the solution of the interface problems with Lagrange multipliers, where the oscillatory part of the solution is resolved by means off smoothing using a new, efficient preconditioner whereas the smooth component of the solution is captured by the collocation-based problem on the auxilliary grid, that is solved directly using a sparse direct solver. A simple adaptive feature is built into the proposed solution method in order to guarantee convergence for ill-conditioned problems. Nmerical results presented for example problems including that of a Boeing crown panel show that the proposed tww-level solution technique outperfrmsnce the standard, single level iterative and direect solvers.     

Efficient numerical schemes for singularly perturbed parabolic initial-boundary-value problems

In this article, we propose two efficient numerical schemes for singularly perturbed parabolic reaction-diffusion initialboundary-value problems. The spatial derivative is replaced by a hybrid scheme, which is a combination of the cubic spline and the classical central difference scheme in both the methods. In the first method, the time derivative is replaced by the Crank-Nicolson scheme, whereas in the second method the time derivative is replaced by the extended-trapezoidal scheme. These schemes are applied on the layer resolving piecewise-uniform Shishkin mesh. Numerical examples show ε -uniform convergence results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of fully discrete schemes with interpolation-type fractional formulas for distributed-order subdiffusion equations

Two fully discrete methods are investigated for simulating the distributed-order sub-diffusion equation in Caputo’s form. The fractional Caputo derivative is approximated by the Caputo’s BDF1 (called L1 early) and BDF2 (or L1-2 when it was first introduced) approximations, which are constructed by piecewise linear and quadratic interpolating polynomials, respectively. It is shown that the first scheme, using the BDF1 formula, possesses the discrete minimum-maximum principle and nonnegativity preservation property such that it is stable and convergent in the maximum norm. The method using the BDF2 formula is shown to be stable and convergent in the discrete H ¹ norm by using the discrete energy method. For problems of distributed order within a certain region, the method is also proven to preserve the discrete maximum principle and nonnegativity property. Extensive numerical experiments are provided to show the effectiveness of numerical schemes, and to examine the initial singularity of the solution. The applicability of our numerical algorithms to a problem with solution which lacks the smoothness near the initial time is examined by employing a class of power-type nonuniform meshes.     

A fully discrete numerical scheme for weighted mean curvature flow

Summary. We analyze a fully discrete numerical scheme approximating the evolution of n–dimensional graphs under anisotropic mean curvature. The highly nonlinear problem is discretized by piecewise linear finite elements in space and semi–implicitly in time. The scheme is unconditionally stable und we obtain optimal error estimates in natural norms. We also present numerical examples which confirm our theoretical results. Received October 2, 2000 / Published online July 25, 2001     

Stability and convergence of fully discrete finite element schemes for the acoustic wave equation

In this paper, we investigate the stability and convergence of some fully discrete finite element schemes for solving the acoustic wave equation where a discontinuous Galerkin discretization in space is used. We first review and compare conventional time-stepping methods for solving the acoustic wave equation. We identify their main properties and investigate their relationship. The study includes the Newmark algorithm which has been used extensively in applications. We present a rigorous stability analysis based on the energy method and derive sharp stability results covering some well-known CFL conditions. A convergence analysis is carried out and optimal a priori error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L ²-norm error over a finite time interval converges optimally as O(h ^p+1+??t ^s ), where p denotes the polynomial degree, s=1 or 2, h the mesh size, and ??t the time step.     

Towards optimal Runge-Kutta smoothers for multigrid methods for unsteady flow problems

We consider Runge-Kutta smoothers in a dual time stepping multigrid method for unsteady flow problems. These smoothers are easily parallelizable and Jacobian-free, making them very attractive for 3D calculations. Existing methods have been designed for steady flows, leading to slow convergence for unsteady problems. Here we determine the free parameters of the smoother to provide optimal damping for high frequency components for the unsteady linear advection equation. This is compared with an RK smoother designed for steady state problems, as commonly used in CFD codes. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Error estimate of a fully discrete defect correction finite element method for unsteady incompressible Magnetohydrodynamics equations

In this study, a fully discrete defect correction finite element method for the unsteady incompressible Magnetohydrodynamics (MHD) equations, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. It is a continuous work of our formal paper [Math Method Appl Sci. 2017. DOI:10.1002/mma.4296]. The defect correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Firstly, the nonlinear MHD equation is solved with an artificial viscosity term. Then, the numerical solutions are improved on the same grid by a linearized defect-correction technique. Then, we introduce the numerical analysis including stability analysis and error analysis. The numerical analysis proves that our method is stable and has an optimal convergence rate. Some numerical results [see Math Method Appl Sci. 2017. DOI:10.1002/mma.4296] show that this method is highly efficient for the unsteady incompressible MHD problems.     

Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems

In this paper, we propose and analyze a fully discrete local discontinuous Galerkin (LDG) finite element method for time-fractional fourth-order problems. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. Stability is ensured by a careful choice of interface numerical fluxes. We prove that our scheme is unconditional stable and convergent. Numerical examples are shown to illustrate the efficiency and accuracy of our scheme.     

Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems

We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of and the higher order spaces, and , with optimal orders of convergence.

     

Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems

We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.     

Difference schemes for fully nonlinear pseudo-hyperbolic systems

The general difference schemes for the first boundary problem of the fully nonlinear pseudo-hyperbolic systemsf(x, t, u,u_x,u_(xx),u_t,u_(tt),u_(xt),u_(xxt))=0are considered in the rectangular domain Q_T={0≤x≤1, 0≤t≤T}, where u(x,t)and f(x, t, u,p_1, p_2, r_1,r_2,q_1,q_2) are two m-dimensional vector functions with m≥1 for(x, t)∈Q_T and u,p_1,p_2,r_1,r_2,q_1,q_2∈R. The existence and the estimates of solutionsfor the finite difference system are established by the fixed point technique. The absolute andrelative stability and convergence of difference schemes are justified by means of a series of a prioriestimates. In the present study, the existence of unique smooth solution of the original problemis assumed. The similar results for nonlinear and quasilinear pseudo-hyperbolic systems are alsoobtained.     

Kinetic energy preserving DG schemes based on summation-by-parts operators on interior node distributions

In this work, a kinetic energy preserving DG scheme in one space dimension using Gauss-Legendre nodes is presented. Stability problems will be demonstrated when using interface terms that are derived from the Lobatto nodes within the Gauss-Legendre skew-symmetric DG formulation. However, combined with correct interface terms, the skew-symmetric DG scheme constructed on Gauss-Legendre nodes is expected to yield higher accuracy compared to its Gauss-Lobatto counterpart. This advantage is demonstrated in the case of viscous compressible flow computation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The stream function coordinate system for solution of one-dimensional unsteady compressible flow problems

A one-dimensional unsteady compressible isentropic flow problem is solved using a floating grid finite difference method. GENMIX, a computer program for solution of two-dimensional parabolic flow problems, has been adopted for this purpose. The advantages of utilizing the staggered floating grid method are demonstrated through solution of flow in a shocktube. The present method is able to precisely locate the discontinuities in temperature and density profiles.     

A note on L^{\infty } bounds and convergence rates of summation-by-parts schemes

In this note, we prove that Summation-by-parts finite difference schemes approximating linear initial-boundary-value problems are stable in \(L^{\infty }\) . This, along with the standard \(L^2\) bound is sufficient for optimal convergence rates.     

Efficient splitting schemes for magneto-hydrodynamic equations

We present in this paper several efficient numerical schemes for the magneto-hydrodynamic (MHD) equations. These semi-discretized (in time) schemes are based on the standard and rotational pressure-correction schemes for the Navier-Stokes equations and do not involve a projection step for the magnetic field. We show that these schemes are unconditionally energy stable, present an effective algorithm for their fully discrete versions and carry out demonstrative numerical experiments.     

Flux-splitting schemes for parabolic problems

Splitting with respect to space variables can be used in solving boundary value problems for second-order parabolic equations. Classical alternating direction methods and locally one-dimensional schemes could be examples of this approach. For problems with rapidly varying coefficients, a convenient tool is the use of fluxes (directional derivatives) as independent variables. The original equation is written as a system in which not only the desired solution but also directional derivatives (fluxes) are unknowns. In this paper, locally one-dimensional additional schemes (splitting schemes) for second-order parabolic equations are examined. By writing the original equation in flux variables, certain two-level locally one-dimensional schemes are derived. The unconditional stability of locally one-dimensional flux schemes of the first and second approximation order with respect to time is proved.     

Explicit-implicit schemes for convection-diffusion-reaction problems

Boundary value problems for time-dependent convection-diffusion-reaction equations are basic models of problems in continuum mechanics. To study these problems, various numerical methods are used. With a finite difference, finite element, or finite volume approximation in space, we arrive at a Cauchy problem for systems of ordinary differential equations whose operator is asymmetric and indefinite. Explicit-implicit approximations in time are conventionally used to construct splitting schemes in terms of physical processes with separation of convection, diffusion, and reaction processes. In this paper, unconditionally stable schemes for unsteady convection-diffusion-reaction equations are constructed with explicit-implicit approximations used in splitting the operator reaction. The schemes are illustrated by a model 2D problem in a rectangle.     

Efficient solver for swirling flow problems in ducts with variable radius

The paper presents preliminary results of a work in progress addressing the hydrodynamic stability of swirling flows problems in ducts with variable radius which imply mathematical modeling, dynamic and stability investigations. The proposed quasi-analytical method aims to obtain the velocity profiles with a low order approximation method of which the computation costs were neglijable and regain the central stagnation zone developed in the fluid. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The discrete picard condition for discrete ill-posed problems

We investigate the approximation properties of regularized solutions to discrete ill-posed least squares problems. A necessary condition for obtaining good regularized solutions is that the Fourier coefficients of the right-hand side, when expressed in terms of the generalized SVD associated with the regularization problem, on the average decay to zero faster than the generalized singular values. This is the discrete Picard condition. We illustrate the importance of this condition theoretically as well as experimentally.This work was carried out during a visit to Dept. of Mathematics, UCLA, and was supported by the Danish Natural Science Foundation, by the National Science Foundation under contract NSF-DMS87-14612, and by the Army Research Office under contract No. DAAL03-88-K-0085.