scholarly journals Convergence rates in parabolic homogenization with time-dependent periodic coefficients

2017 ◽  
Vol 272 (5) ◽  
pp. 2092-2113 ◽  
Author(s):  
Jun Geng ◽  
Zhongwei Shen
Keyword(s):  
Convergence Rates ◽  
Time Dependent ◽  
Periodic Coefficients

scholarly journals Discontinuous Galerkin time discretization methods for parabolic problems with linear constraints

2019 ◽  
Vol 27 (3) ◽  
pp. 155-182 ◽  
Author(s):  
Igor Voulis ◽  
Arnold Reusken
Keyword(s):  
Discontinuous Galerkin ◽  
Error Bounds ◽  
Convergence Rates ◽  
Time Discretization ◽  
Linear Constraints ◽  
Dirichlet Boundary Conditions ◽  
Time Dependent ◽  
Parabolic Problems ◽  
Discretization Methods ◽  
Optimal Convergence

Abstract We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time-dependent) Dirichlet boundary conditions and the time-dependent Stokes equation with an inhomogeneous divergence constraint. Two common ways of treating such linear constraints, namely explicit or implicit (via Lagrange multipliers) are studied. These different treatments lead to different variational formulations of the parabolic problem. For these formulations we introduce a modification of the standard discontinuous Galerkin (DG) time discretization method in which an appropriate projection is used in the discretization of the constraint. For these discretizations (optimal) error bounds, including superconvergence results, are derived. Discretization error bounds for the Lagrange multiplier are presented. Results of experiments confirm the theoretically predicted optimal convergence rates and show that without the modification the (standard) DG method has sub-optimal convergence behavior.

scholarly journals Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 339 ◽  
Author(s):  
Constantin Buşe ◽  
Donal O’Regan ◽  
Olivia Saierli
Keyword(s):  
Positive Integer ◽  
Unit Circle ◽  
Nonnegative Integer ◽  
Monodromy Matrix ◽  
Time Dependent ◽  
Linear Recurrence ◽  
Ulam Stability ◽  
Periodic Coefficients ◽  
Periodic Sequences

Let q ≥ 2 be a positive integer and let ( a j ) , ( b j ) and ( c j ) (with j nonnegative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q − 1 ⋯ A 0 , where A j is defined below. Assume that the eigenvalues x , y , z of the “monodromy matrix” A ( q ) verify the condition ( x − y ) ( y − z ) ( z − x ) ≠ 0 . We prove that the linear recurrence in C x n + 3 = a n x n + 2 + b n x n + 1 + c n x n , n ∈ Z + is Hyers–Ulam stable if and only if ( | x | − 1 ) ( | y | − 1 ) ( | z | − 1 ) ≠ 0 , i.e., the spectrum of A ( q ) does not intersect the unit circle Γ : = { w ∈ C : | w | = 1 } .

scholarly journals Randomized exponential integrators for modulated nonlinear Schrödinger equations

2020 ◽  
Vol 40 (4) ◽  
pp. 2143-2162
Author(s):  
Martina Hofmanová ◽  
Marvin Knöller ◽  
Katharina Schratz
Keyword(s):  
Monte Carlo ◽  
Convergence Rates ◽  
Order Reduction ◽  
Time Dependent ◽  
New Approach ◽  
Nonlinear Schrödinger ◽  
Monte Carlo Approximation ◽  
Exponential Integrator ◽  
Discretization Technique ◽  
Error Behavior

Abstract We consider the nonlinear Schrödinger equation with dispersion modulated by a (formal) derivative of a time-dependent function with fractional Sobolev regularity of class $W^{\alpha ,2}$ for some $\alpha \in (0,1)$. Due to the loss of smoothness in the problem, classical numerical methods face severe order reduction. In this work, we develop and analyze a new randomized exponential integrator based on a stratified Monte Carlo approximation. The new discretization technique averages the high oscillations in the solution allowing for improved convergence rates of order $\alpha +1/2$. In addition, the new approach allows us to treat a far more general class of modulations than the available literature. Numerical results underline our theoretical findings and show the favorable error behavior of our new scheme compared to classical methods.

Invariant curves and time-dependent potentials

1991 ◽  
Vol 11 (2) ◽  
pp. 365-378 ◽  
Author(s):  
Stephane Laederich ◽  
Mark Lev
Keyword(s):  
Hamiltonian Systems ◽  
Time Dependent ◽  
Invariant Curves ◽  
Periodic Coefficients ◽  
Polynomial Potential ◽  
The Stability ◽  
Image Position ◽  
Limiting Case

AbstractIn this paper we prove the existence of invariant curves and thus stability for all time for a class of Hamiltonian systems with time-dependent potentials, namely, for systems of the formwhereis a superquadratic polynomial potential with periodic coefficients. As a limiting case, a proof of the stability of Ulam's problem of a particle bouncing between two periodicially moving walls is given.

scholarly journals VORTEX LIQUIDS AND THE GINZBURG–LANDAU EQUATION

2014 ◽  
Vol 2 ◽  
Author(s):  
MATTHIAS KURZKE ◽  
DANIEL SPIRN
Keyword(s):  
Vortex Dynamics ◽  
Hydrodynamic Limit ◽  
Convergence Rates ◽  
Landau Equation ◽  
Main Tool ◽  
Neumann Boundary ◽  
Time Dependent ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Large Numbers

AbstractWe establish vortex dynamics for the time-dependent Ginzburg–Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quantitative bounds on several fundamental quantities, including the kinetic energy, that lead to explicit convergence rates. For dilute vortex liquids, we prove that sequences of solutions converge to the hydrodynamic limit.

scholarly journals A ROBUST SECOND-ORDER MULTIPLE BALANCE METHOD FOR TIME-DEPENDENT NEUTRON TRANSPORT SIMULATIONS

2021 ◽  
Vol 247 ◽  
pp. 03024
Author(s):  
Ilham Variansyah ◽  
Edward W. Larsen ◽  
William R. Martin
Keyword(s):  
Balance Equation ◽  
Convergence Rates ◽  
Neutron Transport ◽  
Second Order ◽  
Time Dependent ◽  
Discrete Ordinates ◽  
Auxiliary Equation ◽  
Time Step ◽  
Coupled Equations ◽  
Backward Euler

A second-order “Time-Dependent Multiple Balance” (TDMB) method for solving neutron transport problems is introduced and investigated. TDMB consists of solving two coupled equations: (i) the original balance equation (the transport equation integrated over a time step) and (ii) the “balance-like” auxiliary equation (an approximate neutron balance equation). Simple analysis shows that TDMB is second-order accurate and robust (unconditionally free from spurious oscillation). A source iteration (SI) method with diffusion synthetic acceleration (DSA) is formulated to solve these equations. A Fourier analysis reveals that the convergence rates of the proposed iteration schemes for TDMB are similar to those of the common (SI + DSA) schemes for Backward Euler (BE); however, TDMB requires about twice the computational effort per iteration. To demonstrate the theory—accuracy, robustness, and convergence rate—and investigate the efficiency of TDMB, we present results from a discrete ordinates (Sn) research code. Results are discussed, and future work is proposed.

scholarly journals Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 512 ◽  
Author(s):  
Constantin Buşe ◽  
Donal O’Regan ◽  
Olivia Saierli
Keyword(s):  
Positive Integer ◽  
Unit Circle ◽  
Monodromy Matrix ◽  
Time Dependent ◽  
Multiple Eigenvalue ◽  
Linear Recurrence ◽  
Ulam Stability ◽  
Periodic Coefficients ◽  
Periodic Sequences ◽  
Linear Scalar

Let q ≥ 2 be a positive integer and let ( a j ) , ( b j ) , and ( c j ) (with j a non-negative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q − 1 ⋯ A 0 , where A j is as is given below. Assuming that the “monodromy matrix” A ( q ) has at least one multiple eigenvalue, we prove that the linear scalar recurrence x n + 3 = a n x n + 2 + b n x n + 1 + c n x n , n ∈ Z + is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle Γ : = { w ∈ C : | w | = 1 } . Connecting this result with a recently obtained one it follows that the above linear recurrence is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle.

Numerical estimates for the regularization of nonautonomous ill-posed problems

2015 ◽  
Vol 06 (01) ◽  
pp. 1550008 ◽  
Author(s):  
Matthew A. Fury
Keyword(s):  
Diffusion Coefficient ◽  
Initial Value Problems ◽  
Continuous Dependence ◽  
Convergence Rates ◽  
Time Dependent ◽  
Initial Value ◽  
Direct Computation ◽  
Ill Posed ◽  
Well Posed ◽  
Continuous Dependence Of Solutions

The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data. Because direct computation of the solution becomes difficult in this situation, many authors have alternatively approximated the solution by the solution of a closely defined well posed problem. In this paper, we demonstrate this process of regularization for nonautonomous ill-posed problems including the backward heat equation with a time-dependent diffusion coefficient. In the process, we provide two different approximate well posed models and numerically compare convergence rates of their solutions to a known solution of the original ill-posed problem.

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