Explicit Linear Multistep Methods Research Articles

High order modal Discontinuous Galerkin Implicit–Explicit Runge Kutta and Linear Multistep schemes for the Boltzmann model on general polygonal meshes

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 extremely rarefied to dense fluids, with CFL conditions only driven by the hyperbolic transport term. To that aim, we develop modal Discontinuous Galerkin (DG) Implicit–Explicit Runge Kutta schemes (DG-IMEX-RK) and Implicit–Explicit Linear Multistep Methods based on Backward-Finite-Differences (DG-IMEX-BDF) for solving the Boltzmann model on multidimensional unstructured meshes. The solution of the Boltzmann collision operator is obtained through fast spectral methods, while the transport term in the governing equations is discretized relying on an explicit shock-capturing DG method on polygonal tessellations in the physical space. A novel class of WENO-type limiters, based on a shifting of the moments of inertia for each zone of the mesh, is used to control spurious oscillations of the DG solution across discontinuities. The use of Linear Multistep Methods (LMM) allows the Boltzmann solutions to be consistent not only with the compressible Euler limit but also with the Navier–Stokes asymptotic regime. In addition, as numerically proven, they also permit to strongly reduce the computational effort compared to Runge–Kutta approaches while maintaining the same or even larger accuracy. The performances of these different time discretization techniques are measured comparing both precision and efficiency. At the same time, comparisons against simpler relaxation type kinetic models such as the BGK model are proposed. The order of convergence is numerically measured for different regimes and found to agree with the theoretical findings. The new methods are validated considering two-dimensional benchmark test cases typically used in the fluid dynamics community. A prototype engineering problem consisting of a supersonic flow around a NACA 0012 airfoil with space–time-dependent boundary conditions is also presented for which the pressure coefficients are measured.

Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers

Developing new and efficient numerical integration techniques is of great importance in applied mathematics and computer science. Among the variety of available methods, multistep ODE solvers are broadly used in simulation software. Recently, semi-implicit integration proved to be an efficient compromise between implicit and explicit ODE solvers, and multiple high-performance semi-implicit methods were proposed. However, the computational efficiency of any ODE solver can be significantly increased through the introduction of an adaptive integration stepsize, but it requires the estimation of local truncation error. It is known that recently proposed extrapolation semi-implicit multistep methods (ESIMM) cannot operate with existing local truncation error (LTE) estimators, e.g., embedded methods approach, due to their specific right-hand side calculation algorithm. In this paper, we propose two different techniques for local truncation error estimation and study the performance of ESIMM methods with adaptive stepsize control. The first considered approach is based on two parallel semi-implicit solutions with different commutation orders. The second estimator, called the “double extrapolation” method, is a modification of the embedded method approach. The introduction of the double extrapolation LTE estimator allowed us to additionally increase the precision of the ESIMM solver. Using several known nonlinear systems, including stiff van der Pol oscillator, as the testbench, we explicitly show that ESIMM solvers can outperform both implicit and explicit linear multistep methods when implemented with an adaptive stepsize.

Discrete-time nonlinear optimization via zeroing neural dynamics based on explicit linear multi-step methods for tracking control of robot manipulators

In this paper, discrete-time nonlinear optimization (DTNO) for tracking control of robot manipulators is investigated. By utilizing zeroing neural dynamics (ZND) method, a continuous-time ZND (CTZND) model is first proposed for solving the corresponding continuous-time nonlinear optimization (CTNO) problem. Afterwards, three different explicit linear multi-step methods (i.e., explicit linear 3-step, 2-step, and 1-step methods) are respectively presented and investigated. To solve such a DTNO problem, the explicit linear 3-step method is adopted to combine with the CTZND model, and hence a 3-step discrete-time ZND (DTZND) model is proposed. For comparison purposes, 2-step and 1-step DTZND models are also developed. Besides, theoretical analyses indicate the validity and superiority of the proposed 3-step DTZND model. Finally, the numerical experimental results based on a 2-joint robot manipulator and a PUMA560 robot manipulator further verify that the proposed 3-step DTZND model is much superior to the other two DTZND models.

Semi-Implicit Multistep Extrapolation ODE Solvers

Multistep methods for the numerical solution of ordinary differential equations are an important class of applied mathematical techniques. This paper is motivated by recently reported advances in semi-implicit numerical integration methods, multistep and extrapolation solvers. Here we propose a novel type of multistep extrapolation method for solving ODEs based on the semi-implicit basic method of order 2. Considering several chaotic systems and van der Pol nonlinear oscillator as examples, we implemented a performance analysis of the proposed technique in comparison with well-known multistep methods: Adams–Bashforth, Adams–Moulton and the backward differentiation formula. We explicitly show that the multistep semi-implicit methods can outperform the classical linear multistep methods, providing more precision in the solutions for nonlinear differential equations. The analysis of stability regions reveals that the proposed methods are more stable than explicit linear multistep methods. The possible applications of the developed ODE solver are the long-term simulations of chaotic systems and processes, solving moderately stiff differential equations and advanced modeling systems.

Interval versions for special kinds of explicit linear multistep methods

In classical theory of explicit linear multistep methods there are known special kinds of methods which have less function evaluations (in comparison to other multistep methods) and, nevertheless, they give the same accuracy (order) of the approximations obtained. In this paper for such methods we propose their interval versions. It appears that enclosures to the exact solutions obtained by these methods are better in comparison to interval versions of other multistep methods with the same number of steps. The numerical examples presented show that sometimes these enclosures are even better than those obtained by interval methods based on high-order Taylor series.

Strong Stability Preserving Explicit Linear Multistep Methods with Variable Step Size

Strong stability preserving (SSP) methods are designed primarily for time integration of nonlinear hyperbolic PDEs, for which the permissible SSP step size varies from one step to the next. We develop the first SSP linear multistep methods (of order two and three) with variable step size, and prove their optimality, stability, and convergence. The choice of step size for multistep SSP methods is an interesting problem because the allowable step size depends on the SSP coefficient, which in turn depends on the chosen step sizes. The description of the methods includes an optimal step-size strategy. We prove sharp upper bounds on the allowable step size for explicit SSP linear multistep methods and show the existence of methods with arbitrarily high order of accuracy. The effectiveness of the methods is demonstrated through numerical examples.

Two classes of implicit–explicit multistep methods for nonlinear stiff initial-value problems

The initial value problems of nonlinear ordinary differential equations which contain stiff and nonstiff terms often arise from many applications. In order to reduce the computation cost, implicit–explicit (IMEX) methods are often applied to these problems, i.e. the stiff and non-stiff terms are discretized by using implicit and explicit methods, respectively. In this paper, we mainly consider the nonlinear stiff initial-value problems satisfying the one-sided Lipschitz condition and a class of singularly perturbed initial-value problems, and present two classes of the IMEX multistep methods by combining implicit one-leg methods with explicit linear multistep methods and explicit one-leg methods, respectively. The order conditions and the convergence results of these methods are obtained. Some efficient methods are constructed. Some numerical examples are given to verify the validity of the obtained theoretical results.

Estimation of Longest Stability Interval for a Kind of Explicit Linear Multistep Methods

The new explicit linear three‐order four‐step methods with longest interval of absolute stability are proposed. Some numerical experiments are made for comparing different kinds of linear multistep methods. It is shown that the stability intervals of proposed methods can be longer than that of known explicit linear multistep methods.

A technique to construct symmetric variable-stepsize linear multistep methods for second-order systems

Some previous works show that symmetric fixed- and variable-stepsize linear multistep methods for second-order systems which do not have any parasitic root in their first characteristic polynomial give rise to a slow error growth with time when integrating reversible systems. In this paper, we give a technique to construct variable-stepsize symmetric methods from their fixed-stepsize counterparts, in such a way that the former have the same order as the latter. The order and symmetry of the integrators obtained is proved independently of the order of the underlying fixed-stepsize integrators. As this technique looks for efficiency, we concentrate on explicit linear multistep methods, which just make one function evaluation per step, and we offer some numerical comparisons with other one-step adaptive methods which also show a good long-term behaviour.

On the contractivity of implicit–explicit linear multistep methods

This paper is concerned with the class of implicit–explicit linear multistep methods for the numerical solution of initial value problems for ordinary differential equations which are composed of stiff and nonstiff parts. We study the contractivity of such methods, with regard to linear autonomous systems of ordinary differential equations and a (scaled) Euclidean norm. In addition, we derive a strong stability result based on the stability regions of these methods.

Estimations on numerically stable step-size for neutral delay differential systems with multiple delays

We derive two estimations of numerically stable step-size for systems of neutral delay differential equations with multiple delays. The stable step-size for numerical integration of NDDEs with multiple delays can be easily selected by means of the logarithmic norm and the spectral radius of certain matrices. Both explicit linear multistep methods and explicit Runge-Kutta methods are considered.

Contractivity preserving explicit linear multistep methods

We investigate contractivity properties of explicit linear multistep methods in the numerical solution of ordinary differential equations. The emphasis is on the general test-equation $$\frac{d}{{dt}}U(t) = AU(t)$$ , whereA is a square matrix of arbitrary orders?1. The contractivity is analysed with respect to arbitrary norms in thes-dimensional space (which are not necessarily generated by an inner product). For given order and stepnumber we construct optimal multistep methods allowing the use of a maximal stepsize.

Spline-collocation with adaptive mesh grading for solving the stochastic collection equation

The method of collocation using cubic B-splines and an adaptive mesh is applied to the solution of the partial integro-differential equation that describes the continuous mass spectrum of particles undergoing stochastic collection growth. Temporal discretization and a number of explicit and implicit linear multistep methods are employed to solve the system of ordinary differential equations. The numerical method is tested by solving model problems that describe the coalescence of particles for both single and double initial distributions. It is found that accurate solutions can be obtained using a small number of B-splines, and that adaptive mesh grading improves the accuracy of the solutions without needing additional nodal points.

A Special Class of Explicit Linear Multistep Methods as Basic Methods for the Correction in the Dominant Space Technique

A class of explicit linear multistep methods is suggested as basic methods for the CDS schemes introduced in [3]. These schemes are designed for the numerical solution of certain stiff ordinary differential equations, and operate with dominant eigenvalues, and the corresponding eigenvectors, of the Jacobian. The motivation, and the stability analysis for CDS schemes assumes that the eigensystem is constant. Here methods are introduced that perform particularly well if the eigensystem is not constant. In a certain sense the methods introduced here can be considered explicit approximations to the well-known implicit backward-differentiation formulas used by Gear [6] for the stiff option of his o.d.e. solver.

A special class of explicit linear multistep methods as basic methods for the correction in the dominant space technique

A class of explicit linear multistep methods is suggested as basic methods for the CDS schemes introduced in [3]. These schemes are designed for the numerical solution of certain stiff ordinary differential equations, and operate with dominant eigenvalues, and the corresponding eigenvectors, of the Jacobian. The motivation, and the stability analysis for CDS schemes assumes that the eigensystem is constant. Here methods are introduced that perform particularly well if the eigensystem is not constant. In a certain sense the methods introduced here can be considered explicit approximations to the well-known implicit backward-differentiation formulas used by Gear [6] for the stiff option of his o.d.e. solver.

Inverse linear multistep methods for the numerical solution of initial value problems of ordinary differential equations

The well-known explicit linear multistep methods for the numerical solution of ordinary differential equations advance the numerical solution from x n + k − 1 {x_{n + k - 1}} to x n + k {x_{n + k}} by computing some numerical approximation from back values and then evaluating the problem defining function to obtain an approximation of the derivative. In this paper similar methods are proposed that first compute an approximation to the derivative and then compute an approximation to the exact solution, either by evaluating a suitable function, or by solving a nonlinear system of equations. The methods can be applied to initial value problems where the exact solution is explicitly given in terms of the derivative. They can also be applied in the context of the CDS technique for certain stiff initial value problems of ordinary differential equations, introduced in [1 ] and [2]. Local accuracy and stability of the methods are defined and investigated, and specific methods, containing free parameters, are given. The methods are not convergent, but they yield very good numerical results if applied to the type of problem they are designed for. Their major advantage is that they significantly reduce the amount of implicitness necessary in the numerical solution of certain problems.

Inverse Linear Multistep Methods for the Numerical Solution of Initial Value Problems of Ordinary Differential Equations

The well-known explicit linear multistep methods for the numerical solution of ordinary differential equations advance the numerical solution from ${x_{n + k - 1}}$ to ${x_{n + k}}$ by computing some numerical approximation from back values and then evaluating the problem defining function to obtain an approximation of the derivative. In this paper similar methods are proposed that first compute an approximation to the derivative and then compute an approximation to the exact solution, either by evaluating a suitable function, or by solving a nonlinear system of equations. The methods can be applied to initial value problems where the exact solution is explicitly given in terms of the derivative. They can also be applied in the context of the CDS technique for certain stiff initial value problems of ordinary differential equations, introduced in [1 ] and [2]. Local accuracy and stability of the methods are defined and investigated, and specific methods, containing free parameters, are given. The methods are not convergent, but they yield very good numerical results if applied to the type of problem they are designed for. Their major advantage is that they significantly reduce the amount of implicitness necessary in the numerical solution of certain problems.