An implicit mass-matrix penalization (IMMP) of Hamiltonian dynamics is proposed, and associated dynamical integrators, as well as sampling Monte-Carlo schemes, are analyzed for systems with multiple time scales. The penalization is based on an extended Hamiltonian with artificial constraints associated with some selected DOFs. The penalty parameters enable arbitrary tuning of timescales for the selected DOFs. The IMMP dynamics is shown to be an interpolation between the exact Hamiltonian dynamics and the dynamics with rigid constraints. This property translates in the associated numerical integrator into a tunable trade-off between stability and dynamical modification. Moreover, a penalty that vanishes with the time-step yields order two convergent schemes for the exact dynamics. Moreover, by construction, the resulting dynamics preserves the canonical equilibrium distribution in position variables, up to a computable geometric correcting potential, leading to Metropolis-like unbiased sampling algorithms. The algorithms can be implemented with a simple modification of standard geometric integrators with algebraic constraints imposed on the selected DOFs, and has no additional complexity in terms of enforcing the constraints and force evaluations. The properties of the IMMP method are demonstrated numerically on the $N$-alkane model, showing that the time-step stability region of integrators and the sampling efficiency can be increased with a gain that grows with the size of the system. This feature is mathematically analyzed for a harmonic atomic chain model. When a large stiffness parameter is introduced, the IMMP method is shown to be asymptotically stable and to converge towards the heuristically expected Markovian effective dynamics on the slow manifold.
We design, analyze, and implement a new conservative Discontinuous Galerkin (DG) method for the simulation of solitary wave solutions to the generalized Korteweg-de Vries (KdV) Equation. The key feature of our method is the conservation, at the numerical level, of the mass, energy and Hamiltonian that are conserved by exact solutions of all KdV equations. To our knowledge, this is the first DG method that conserves all these three quantities, a property critical for the accurate long-time evolution of solitary waves. To achieve the desired conservation properties, our novel idea is to introduc…
High-order Conservative Discontinuous Galerkin Methods via Implicit Penalization for the Generalized Korteweg-de Vries Equation and the Hirota-Satsuma KdV System
University of Massachusetts Dartmouth（马萨诸塞大学达特茅斯分校）
An L-Stable Implicit Two-Stage Fourth-Order Temporal Discretization Scheme for Lax-Wendroff-Type Solvers Applied to Stiff Problems
School of Mathematics and Information Science, Henan Polytechnic University（河南理工大学数学与信息科学学院）；School of Mechatronical Engineering, Beijing Institute of Technology（北京理工大学机械与车辆学院）
Many hyperbolic and kinetic equations contain a non-stiff convection/transport part and a stiff relaxation/collision part (characterized by the relaxation or mean free time $\varepsilon$). To solve this type of problems, implicit-explicit (IMEX) multistep methods have been widely used and their performance is understood well in the non-stiff regime ($\varepsilon=O(1)$) and limiting regime ($\varepsilon\rightarrow 0$). However, in the intermediate regime (say, $\varepsilon=O(Δt)$), uniform accuracy has been reported numerically without a complete theoretical justification (except some asymptoti…
In this work, an approximate family of implicit multiderivative Runge-Kutta (MDRK) time integrators for stiff initial value problems is presented. The approximation procedure is based on the recent Approximate Implicit Taylor method (Baeza et al. in Comput. Appl. Math. 39:304, 2020). As a Taylor method can be written in MDRK format, the novel family constitutes a multistage generalization. Two different alternatives are investigated for the computation of the higher order derivatives: either directly as part of the stage equation, or either as a separate formula for each derivative added on to…
In many applications, the governing PDE to be solved numerically contains a stiff component. When this component is linear, an implicit time stepping method that is unencumbered by stability restrictions is often preferred. On the other hand, if the stiff component is nonlinear, the complexity and cost per step of using an implicit method is heightened, and explicit methods may be preferred for their simplicity and ease of implementation. In this article, we analyze new and existing linearly stabilized schemes for the purpose of integrating stiff nonlinear PDEs in time. These schemes compute t…
Extrapolation-based implicit-explicit general linear methods
Università degli studi di Salerno（萨莱诺大学）；Arizona State University（亚利桑那州立大学）；AGH University of Science and Technology（AGH科技大学）；Virginia Polytechnic Institute & State University（弗吉尼亚理工大学）
We study a class of semi-implicit Taylor-type numerical methods that are easy to implement and designed to solve multidimensional stochastic differential equations driven by a general rough noise, e.g. a fractional Brownian motion. In the multiplicative noise case, the equation is understood as a rough differential equation in the sense of T.~Lyons. We focus on equations for which the drift coefficient may be unbounded and satisfies a one-sided Lipschitz condition only. We prove well-posedness of the methods, provide a full analysis, and deduce their convergence rate. Numerical experiments sho…
In this paper, a new implicit-explicit local method with an arbitrary order is produced for stiff initial value problems. Here, a general method for one-step time integrations has been created, considering a direction free approach for integrations leading to a numerical method with parameter-based stability preservation. Adaptive procedures depending on the problem types for the current method are explained with the help of local error estimates to minimize the computational cost. Priority error analysis of the current method is made, and order conditions are presented in terms of direction p…
In this paper, the design and analysis of high order accurate IMEX finite volume schemes for the compressible Euler-Poisson (EP) equations in the quasineutral limit is presented. As the quasineutral limit is singular for the governing equations, the time discretisation is tantamount to achieving an accurate numerical method. To this end, the EP system is viewed as a differential algebraic equation system (DAEs) via the method of lines. As a consequence of this vantage point, high order linearly semi-implicit (SI) time discretisation are realised by employing a novel combination of the direct a…
All-stages-implicit and strong-stability-preserving implicit-explicit Runge-Kutta time discretization schemes for hyperbolic systems with stiff relaxation terms
Institute of Fluid Physics, CAEP（中国工程物理研究院流体物理研究所）；Department of Modern Physics, University of Science and Technology of China（中国科学技术大学近代物理系）
Unconditionally stable implicit time-marching methods are powerful in solving stiff differential equations efficiently. In this work, a novel framework to handle stiff physical terms implicitly is proposed. Both physical and numerical stiffness originating from convection, diffusion and source terms (typically related to reaction) can be handled by a set of predefined Time-Accurate and highly-Stable Explicit (TASE) operators in a unified framework. The proposed TASE operators act as preconditioners on the stiff terms and can be deployed to any existing explicit time-marching methods straightfo…
Efficient high order schemes for stiff ODEs in cardiac electrophysiology
INRIA Bordeaux Sud Ouest, Université de Bordeaux（法国国家信息与自动化研究所波尔多西南分部，波尔多大学）；Institut de Mathématiques de Bordeaux, UMR CNRS 5241（波尔多数学研究所）；Laboratoire de Mathématiques et de leurs Applications, UMR CNRS 5142, Université de Pau et des Pays de l’Adour（波尔多大学及阿杜尔地区大学数学及其应用实验室）
We present a new linearly implicit exponential integrator that preserves the polynomial first integrals or Lyapunov functions for the conservative and dissipative stiff equations, respectively. The method is tested by both oscillated ordinary differential equations and partial differential equations, e.g., an averaged system in wind-induced oscillation, the Fermi-Pasta-Ulam systems, and the polynomial pendulum oscillators. The numerical simulations confirm the conservative properties of the proposed method and demonstrate its good behavior in superior running speed when compared with fully imp…
Many interesting applications of hyperbolic systems of equations are stiff, and require the time step to satisfy restrictive stability conditions. One way to avoid small time steps is to use implicit time integration. Implicit integration is quite straightforward for first order schemes. High order schemes instead need also to control spurious oscillations, which requires limiting in space and time also in the implicit case. We propose a framework to simplify considerably the application of high order non oscillatory schemes through the introduction of a low order implicit predictor, which is …
