Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRobust Convergence of Parareal Algorithms with Arbitrarily High-order Fine Propagators
78
0
0.0
(
0
)
Added by
Zhi Zhou
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The aim of this paper is to analyze the robust convergence of a class of parareal algorithms for solving parabolic problems. The coarse propagator is fixed to the backward Euler method and the fine propagator is a high-order single step integrator. Under some conditions on the fine propagator, we show that there exists some critical $J_*$ such that the parareal solver converges linearly with a convergence rate near $0.3$, provided that the ratio between the coarse time step and fine time step named $J$ satisfies $J ge J_*$. The convergence is robust even if the problem data is nonsmooth and incompatible with boundary conditions. The qualified methods include all absolutely stable single step methods, whose stability function satisfies $|r(-infty)|<1$, and hence the fine propagator could be arbitrarily high-order. Moreover, we examine some popular high-order single step methods, e.g., two-, three- and four-stage Lobatto IIIC methods, and verify that the corresponding parareal algorithms converge linearly with a factor $0.31$ and the threshold for these cases is $J_* = 2$. Intensive numerical examples are presented to support and complete our theoretical predictions.
rate research
Read More
We present a paradigm for developing arbitrarily high order, linear, unconditionally energy stable numerical algorithms for gradient flow models. We apply the energy quadratization (EQ) technique to reformulate the general gradient flow model into an equivalent gradient flow model with a quadratic free energy and a modified mobility. Given solutions up to $t_n=n Delta t$ with $Delta t$ the time step size, we linearize the EQ-reformulated gradient flow model in $(t_n, t_{n+1}]$ by extrapolation. Then we employ an algebraically stable Runge-Kutta method to discretize the linearized model in $(t_n, t_{n+1}]$. Then we use the Fourier pseudo-spectral method for the spatial discretization to match the order of accuracy in time. The resulting fully discrete scheme is linear, unconditionally energy stable, uniquely solvable, and may reach arbitrarily high order. Furthermore, we present a family of linear schemes based on prediction-correction methods to complement the new linear schemes. Some benchmark numerical examples are given to demonstrate the accuracy and efficiency of the schemes.
We develop and analyze a class of maximum bound preserving schemes for approximately solving Allen--Cahn equations. We apply a $k$th-order single-step scheme in time (where the nonlinear term is linearized by multi-step extrapolation), and a lumped mass finite element method in space with piecewise $r$th-order polynomials and Gauss--Lobatto quadrature. At each time level, a cut-off post-processing is proposed to eliminate extra values violating the maximum bound principle at the finite element nodal points. As a result, the numerical solution satisfies the maximum bound principle (at all nodal points), and the optimal error bound $O(tau^k+h^{r+1})$ is theoretically proved for a certain class of schemes. These time stepping schemes under consideration includes algebraically stable collocation-type methods, which could be arbitrarily high-order in both space and time. Moreover, combining the cut-off strategy with the scalar auxiliary value (SAV) technique, we develop a class of energy-stable and maximum bound preserving schemes, which is arbitrarily high-order in time. Numerical results are provided to illustrate the accuracy of the proposed method.
This paper proposes a new class of arbitarily high-order conservative numerical schemes for the generalized Korteweg-de Vries (KdV) equation. This approach is based on the scalar auxiliary variable (SAV) method. The equation is reformulated into an equivalent system by introducing a scalar auxiliary variable, and the energy is reformulated into a sum of two quadratic terms. Therefore, the quadratic preserving Runge-Kutta method will preserve both the mass and the reformulated energy in the discrete time flow. With the Fourier pseudo-spectral spatial discretization, the scheme conserves the first and third invariant quantities (momentum and energy) exactly in the fully discrete sense. The discrete mass possesses the precision of the spectral accuracy.
We present a systematical approach to developing arbitrarily high order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation laws. Utilizing the energy quadratization (EQ) method, We formulate the gradient flow model into an equivalent form with a corresponding quadratic free energy functional. Based on the equivalent form with a quadratic energy, we propose two classes of energy stable numerical approximations. In the first approach, we use a prediction-correction strategy to improve the accuracy of linear numerical schemes. In the second approach, we adopt the Gaussian collocation method to discretize the equivalent form with a quadratic energy, arriving at an arbitrarily high-order scheme for gradient flow models. Schemes derived using both approaches are proved rigorously to be unconditionally energy stable. The proposed schemes are then implemented in four gradient flow models numerically to demonstrate their accuracy and effectiveness. Detailed numerical comparisons among these schemes are carried out as well. These numerical strategies are rather general so that they can be readily generalized to solve any thermodynamically consistent PDE models.
We design a Hybrid High-Order (HHO) scheme for the Poisson problem that is fully robust on polytopal meshes in the presence of small edges/faces. We state general assumptions on the stabilisation terms involved in the scheme, under which optimal error estimates (in discrete and continuous energy norms, as well as $L^2$-norm) are established with multiplicative constants that do not depend on the maximum number of faces in each element, or the relative size between an element and its faces. We illustrate the error estimates through numerical simulations in 2D and 3D on meshes designed by agglomeration techniques (such meshes naturally have elements with a very large numbers of faces, and very small faces).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
