In this note we propose and analyze novel implicit-explicit methods based on second order strong stability preserving multistep time discretizations. Several schemes are developed, and a linear stability analysis is performed to study their properties with respect to the implicit and explicit eigenvalues. One of the proposed schemesis found to have very good stability properties, with implicit A-stability for the entire explicit stability domain. The properties of the other proposed schemes are comparable to those of traditional methods found in the literature.
We consider the construction of semi-implicit linear multistep methods which can be applied to time dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino, Filbet and Russo (2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allows, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work we develop a general setting for the construction of high order semi-implicit linear multistep methods and ana…
We consider the development of high order space and time numerical methods based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic systems with relaxation. More specifically, we consider hyperbolic balance laws in which the convection and the source term may have very different time and space scales. As a consequence the nature of the asymptotic limit changes completely, passing from a hyperbolic to a parabolic system. From the computational point of view, standard numerical methods designed for the fluid-dynamic scaling of hyperbolic systems with relaxation present several…
Time integration methods for solving initial value problems are an important component of many scientific and engineering simulations. Implicit time integrators are desirable for their stability properties, significantly relaxing restrictions on timestep size. However, implicit methods require solutions to one or more systems of nonlinear equations at each timestep, which for large simulations can be prohibitively expensive. This paper introduces a new family of linearly implicit multistep methods (LIMM), which only requires the solution of one linear system per timestep. Order conditions and …
A finite element discretization with semi-implicit nonlinear multistep scheme for a two-dimensional competition-diffusion system of three competing species with different mobility rates
College of Integrative Sciences and Arts, Arizona State University（亚利桑那州立大学综合科学与艺术学院）；Department of Mathematics, Jimma University（吉马大学数学系）；Department of Mathematics and Applied Mathematics, University of the Western Cape（西开普大学数学与应用数学系）
We consider a finite element method with symmetric stabilisation for the discretisation of the transient convection--diffusion equation. For the time-discretisation we consider either the second order backwards differentiation formula or the Crank-Nicolson method. Both the convection term and the associated stabilisation are treated explicitly using an extrapolated approximate solution. We prove stability of the method and the $τ^2 + h^{p+{\frac12}}$ error estimates for the $L^2$-norm under either the standard hyperbolic CFL condition, when piecewise affine ($p=1$) approximation is used, or in…
Stability analysis of consistent splitting implicit-explicit multistep methods up to ninth-order accuracy for incompressible flows
School of Mathematics, Nanjing University of Aeronautics and Astronautics（南京航空航天大学数学学院）；Key Laboratory of Mathematical Modeling and High Performance Computing of Air Vehicles (NUAA), MIIT（航空飞行器数学建模与高性能计算教育部重点实验室（南航））；School of Mathematics, Nanjing Audit University（南京审计大学数学学院）
Energy Dissipation Analysis of Implicit-Explicit Linear Multistep Methods for Gradient Flows Using General Multipliers
The Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））；Shenzhen International Center for Industrial and Applied Mathematics, Shenzhen Research Institute of Big Data（深圳大数据研究院深圳市工业与应用数学中心）；Xiamen University（厦门大学）
Deterministic solutions of the Boltzmann equation represent a real challenge due to the enormous computational effort which is required to produce such simulations and often stochastic methods such as Direct Simulation Monte Carlo (DSMC) are used instead due to their lower computational cost. In this work, we show that combining different technologies for the discretization of the velocity space and of the physical space coupled with suitable time integration techniques, it is possible to compute very precise deterministic approximate solutions of the Boltzmann model in different regimes, from…
A semi-generating function approach to the stability of implicit-explicit multistep methods for nonlinear parabolic equations
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Key Laboratory of Mathematical Modeling and High Performance Computing of Air Vehicles (NUAA), MIIT（航空飞行器数学建模与高性能计算重点实验室（工信部））；The Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））；Shenzhen International Center for Industrial and Applied Mathematics, Shenzhen Research Institute of Big Data（深圳市大数据研究院工业与应用数学国际中心）；Guangzhou Nanfang College（广州南方学院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
In this paper, a stabilized second order in time accurate linear exponential time differencing (ETD) scheme for the no-slope-selection thin film growth model is presented. An artificial stabilizing term $Aτ^2\frac{\partialΔ^2 u}{\partial t}$ is added to the physical model to achieve energy stability, with ETD-based multi-step approximations and Fourier collocation spectral method applied in the time integral and spatial discretization of the evolution equation, respectively. Long time energy stability and detailed $\ell^{\infty}(0,T; \ell^2)$ error analysis are provided based on the energy met…
King Abdullah University of Science & Technology（阿卜杜拉国王科技大学）；Széchenyi István University（塞切尼伊什特万大学）；University of Massachusetts Dartmouth（马萨诸塞大学达特茅斯分校）
We introduce a new $\mathbf F$-modulated energy stability framework for general linear multistep methods. We showcase the theory for the two dimensional molecular beam epitaxy model with no slope selection which is a prototypical gradient flow with Lipschitz-bounded nonlinearity. We employ a class of representative BDF$k$, $2\le k \le 5$ discretization schemes with explicit $k^{\mathrm{th}}$-order extrapolation of the nonlinear term. We prove the uniform-in-time boundedness of high Sobolev norms of the numerical solution. The upper bound is unconditional, i.e. regardless of the size of the tim…
The purpose of this work is to introduce a new idea of how to avoid the factorization of large matrices during the solution of stiff systems of ODEs. Starting from the general form of an explicit linear multistep method we suggest to adaptively choose its coefficients on each integration step in order to minimize the norm of the residual of an implicit BDF formula. Thereby we reduce the number of unknowns on each step from $n$ to $O(1)$, where $n$ is the dimension of the ODE system. We call this type of methods Minimal Residual Multistep (MRMS) methods. In the case of linear non-autonomous pro…
We present and investigate a new type of implicit fractional linear multistep method of order two for fractional initial value problems. The method is obtained from the second order super convergence of the Grünwald-Letnikov approximation of the fractional derivative at a non-integer shift point. The proposed method is of order two consistency and coincides with the backward difference method of order two for classical initial value problems when the order of the derivative is one. The weight coefficients of the proposed method are obtained from the Grünwald weights and hence computationally e…
This paper proposes several explicit and implicit multistep frequency response optimized integrators considering first or second order derivative. A prediction-based method aiming at accelerating a novel power system transient simulation scheme without impacting its accuracy is further put forward utilizing the proposed numerical integrators and some others available in the literature. Case studies verify the effectiveness of the proposed prediction method. Although they are utilized to accelerate the simulation scheme in this paper, the proposed numerical integrators are in fact general-purpo…
Most research on preconditioners for time-dependent PDEs has focused on implicit multi-step or diagonally-implicit multi-stage temporal discretizations. In this paper, we consider monolithic multigrid preconditioners for fully-implicit multi-stage Runge-Kutta (RK) time integration methods. These temporal discretizations have very attractive accuracy and stability properties, but they couple the spatial degrees of freedom across multiple time levels, requiring the solution of very large linear systems. We extend the classical Vanka relaxation scheme to implicit RK discretizations of saddle poin…
In this work, we determine the full expression for the global truncation error of hyperbolic partial differential equations (PDEs). In particular, we use theoretical analysis and symbolic algebra to find exact expressions for the coefficients of the generic global truncation error. Our analysis is valid for any hyperbolic PDE, be it linear or non-linear, and employing finite difference, finite volume, or finite element discretization in space, and advanced in time with a predictor-corrector, multistep, or a deferred correction method, belonging to the Method of Lines. Furthermore, we discuss t…
A novel theoretical convergence rate estimate for a Balancing Domain Decomposition by Constraints algorithm is proven for the solution of the cardiac Bidomain model, describing the propagation of the electric impulse in the cardiac tissue. The non-linear system arises from a fully implicit time discretization and a monolithic solution approach. The preconditioned non-symmetric operator is constructed from the linearized system arising within the Newton-Krylov approach for the solution of the non-linear problem; we theoretically analyze and prove a convergence rate bound for the Generalised Min…
An extension of the two-step staggered time discretization of linear elastodynamics in stress-velocity form to systems involving internal variables subjected to a possibly non-linear dissipative evolution is proposed. The original scheme is thus enhanced by another step for the internal variables which, in general, is implicit, although even this step might be explicit if no spatial gradients of the internal variables are involved. Using an abstract Banach-space formulation, a-priori estimates and convergence are proved under a CFL condition. The developed three-step scheme finds applications …
For the Landau--Lifshitz--Gilbert (LLG) equation of micromagnetics we study linearly implicit backward difference formula (BDF) time discretizations up to order $5$ combined with higher-order non-conforming finite element space discretizations, which are based on the weak formulation due to Alouges but use approximate tangent spaces that are defined by $L^2$-averaged instead of nodal orthogonality constraints. We prove stability and optimal-order error bounds in the situation of a sufficiently regular solution. For the BDF methods of orders $3$ to~$5$, this requires %a mild time step restricti…
Conservation properties of iterative methods applied to implicit finite volume discretizations of nonlinear conservation laws are analyzed. It is shown that any consistent multistep or Runge-Kutta method is globally conservative. Further, it is shown that Newton's method, Krylov subspace methods and pseudo-time iterations are globally conservative while the Jacobi and Gauss-Seidel methods are not in general. If pseudo-time iterations using an explicit Runge-Kutta method are applied to a locally conservative discretization, then the resulting scheme is also locally conservative. However, the co…
We provide a framework for high-order discretizations of nonlinear scalar convection-diffusion equations that satisfy a discrete maximum principle. The resulting schemes can have arbitrarily high order accuracy in time and space, and can be stable and maximum-principle-preserving (MPP) with no step size restriction. The schemes are based on a two-tiered limiting strategy, starting with a high-order limiter-based method that may have small oscillations or maximum-principle violations, followed by an additional limiting step that removes these violations while preserving high order accuracy. The…
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）
This study concerns numerical methods for efficiently solving the Richards equation where different weak formulations and computational techniques are analyzed. The spatial discretizations are based on standard or mixed finite element methods. Different implicit and semi-implicit temporal discretization techniques of second-order accuracy are studied. To obtain a linear system for the semi-implicit schemes, we propose second-order techniques using extrapolation formulas and/or semi-implicit Taylor approximations for the temporal discretization of nonlinear terms. A numerical convergence study …
Department of Applied Physics and Applied Mathematics, Columbia University（哥伦比亚大学应用物理与应用数学系）
Linear multistep methods (LMMs) are popular time discretization techniques for the numerical solution of differential equations. Traditionally they are applied to solve for the state given the dynamics (the forward problem), but here we consider their application for learning the dynamics given the state (the inverse problem). This repurposing of LMMs is largely motivated by growing interest in data-driven modeling of dynamics, but the behavior and analysis of LMMs for discovery turn out to be significantly different from the well-known, existing theory for the forward problem. Assuming a high…
Two novel parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are here constructed, analyzed and tested numerically for implicit time discretizations of the three-dimensional Bidomain system of equations. This model represents the most advanced mathematical description of the cardiac bioelectrical activity and it consists of a degenerate system of two non-linear reaction-diffusion partial differential equations (PDEs), coupled with a stiff system of ordinary differential equations (ODEs). A fin…
We analyze, from the viewpoint of positivity preservation, certain discretizations of a fundamental partial differential equation, the one-dimensional advection equation with periodic boundary condition. The full discretization is obtained by coupling a finite difference spatial semi-discretization (the second- and some higher-order centered difference schemes, or the Fourier spectral collocation method) with an arbitrary $θ$-method in time (including the forward and backward Euler methods, and a second-order method by choosing $θ\in [0,1]$ suitably). The full discretization generates a two-pa…
We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which employs a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. The proposed method allows for large time steps, while avoiding the solution of large linear systems, which would be required by an implicit time discretization technique. Numerical experiments demonstrate the effectiveness of the proposed appr…
Variable steps implicit-explicit multistep methods for PDEs have been presented in [17], where the zero-stability is studied for ODEs; however, the stability analysis still remains an open question for PDEs. Based on the idea of linear multistep methods, we present a simple weighted and shifted BDF2 methods with variable steps for the parabolic problems, which serve as a bridge between BDF2 and Crank-Nicolson scheme. The contributions of this paper are as follows: we first prove that the optimal adjacent time-step ratios for the weighted and shifted BDF2, which greatly improve the maximum time…
Invariant domain preserving limiting of time explicit and time implicit discretizations for systems of conservation laws
ONERA, Institut Polytechnique de Paris（法国航空航天研究院，巴黎理工学院）；MONHADE, Equipe Inria-ONERA, DAAA, ONERA, Institut Polytechnique de Paris（MONHADE，Inria-ONERA团队，DAAA，法国航空航天研究院，巴黎理工学院）
