New Class Of Numerical Methods Research Articles

High order linearly implicit methods for evolution equations

This paper introduces a new class of numerical methods for the time integration of evolution equations set as Cauchy problems of ODEs or PDEs. The systematic design of these methods mixes the Runge–Kutta collocation formalism with collocation techniques, in such a way that the methods are linearly implicit and have high order. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to integrate come from the space discretization of evolution PDEs. Moreover, these methods are expected to be efficient since they only require to solve one linear system of equations at each time step, and efficient techniques from the literature can be used to do so. After the introduction of the methods, we set suitable definitions of consistency and stability for these methods. This allows for a proof that arbitrarily high order linearly implicit methods exist and converge when applied to ODEs. Eventually, we perform numerical experiments on ODEs and PDEs that illustrate our theoretical results for ODEs, and compare our methods with standard methods for several evolution PDEs.

A robust, high-order implicit shock tracking method for simulation of complex, high-speed flows

High-order implicit shock tracking (fitting) is a new class of numerical methods to approximate solutions of conservation laws with non-smooth features, e.g., contact lines, shock waves, and rarefactions. These methods align elements of the computational mesh with non-smooth features to represent them perfectly, allowing high-order basis functions to approximate smooth regions of the solution without the need for nonlinear stabilization, which leads to accurate approximations on traditionally coarse meshes. The hallmark of these methods is the underlying optimization formulation whose solution is a feature-aligned mesh and the corresponding high-order approximation to the flow; the key challenge is robustly solving the central optimization problem. In this work, we develop a robust optimization solver for high-order implicit shock tracking methods so they can be reliably used to simulate complex, high-speed, compressible flows in multiple dimensions. The proposed method integrates practical robustness measures into a sequential quadratic programming method that employs a Levenberg-Marquardt approximation of the Hessian and line-search globalization. The robustness measures include dimension- and order-independent simplex element collapses, mesh smoothing, and element-wise solution re-initialization, which prove to be necessary to reliably track complex discontinuity surfaces, such as curved and reflecting shocks, shock formation, and shock-shock interaction. A series of nine numerical experiments—including two- and three-dimensional compressible flows with complex discontinuity surfaces—are used to demonstrate: 1) the robustness of the solver, 2) the meshes produced are high-quality and track continuous, non-smooth features in addition to discontinuities, 3) the method achieves the optimal convergence rate of the underlying discretization even for flows containing discontinuities, and 4) the method produces highly accurate solutions on extremely coarse meshes relative to approaches based on shock capturing.

Numerical schemes for ordinary delay differential equations with random noise

Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) which have a stochastic process in their vector field functions. They have been used in a wide range of applications such as biology, medicine and engineering and play an important role in the theory of random dynamical systems. RODEs can be investigated pathwise as deterministic ODEs, however, the classical numerical methods for ODEs do not attain original order of convergence because the stochastic process has at most Hölder continuous sample paths and the resulting vector is also at most Hölder continuous in time. Recently, Jenzen & Kloeden derived new class of numerical methods for RODEs using integral versions of implicit Taylor-like expansions and developed arbitrary higher order schemes for RODEs. Their idea can be applied to random ordinary delay differential equations (RODDEs) by implementing Taylor-like expansions in the corresponding delay term. In this paper, numerical methods for RODDEs are systematically constructed based on Taylor-like expansions and they are applied to virus dynamics model with random fluctuations and time delay.

An efficient technique to solve nonlinear equations using multiplicative calculus

In this paper, we develop an efficient technique in the framework of multiplicative calculus and suggest a new class of numerical methods for solving multiplicative nonlinear equations \textit{g}(\textit{x}% ) = 1. We also develop the convergence criteria of the proposed methods. We solve the population growth model and minimization problem, which demonstrate the implementation and efficiency of the new techniques. We also show that these techniques perform much better as compared to the similar ordinary methods for solving ordinary nonlinear equations \textit{f}(\textit{x}) = 0.

Volume-preserving algorithms for charged particle dynamics

The paper reports the development of volume-preserving algorithms using the splitting technique for charged particle motion under the Lorentz force. The source-free nature of the Lorentz vector field has been investigated. Based on the volume-preserving property of the dynamics, a class of numerical methods for advancing charged particles in a general electromagnetic field has been constructed by splitting the classical evolution operator. This new class of numerical methods, which includes the Boris algorithm as a special case, conserves phase space volume, and globally bounds the numerical errors of energy, momentum, and other adiabatic invariants up to the order of the method over a very long simulation time. These algorithms can be computed explicitly, and thus are effective for the long-term simulation of the multi-scale dynamics of plasmas.

Implicit difference schemes for quasilinear parabolic functional equations

AbstractWe present a new class of numerical methods for quasilinear parabolic functional differential equations with initial boundary conditions of the Robin type. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show that the new methods are considerable better than the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given functions with respect to functional variables. Results obtained in the paper can be applied to differential equations with deviated variables and to differential integral problems.

Welding Heat Transfer Analysis Using Element Free Galerkin Method

Mesh-less methods belong to a new class of numerical methods in computational mechanics and offer several advantages over the conventional mesh-based methods. They enable modelling of processes involving high deformation, severe discontinuities (e.g. fracture) and multiple physical processes. These types of situations are usually encountered in arc welding, rendering its modelling suitable via mesh-less methods. In this paper, a mesh-less Element Free Galerkin (EFG) method has been developed to model the heat transfer during welding. The results predicted by the EFG method are found to be in close agreement with those obtained by the finite element method and those observed in welding experiments. This demonstrates the effectiveness and utilities of the EFG method for modelling and understanding the heat transfer processes in arc welding.

Microscopically implicit–macroscopically explicit schemes for the BGK equation

In this work a new class of numerical methods for the BGK model of kinetic equations is introduced. The schemes proposed are implicit with respect to the distribution function, while the macroscopic moments are evolved explicitly. In this fashion, the stability condition on the time step coincides with a macroscopic CFL, evaluated using estimated values for the macroscopic velocity and sound speed. Thus the stability restriction does not depend on the relaxation time and it does not depend on the microscopic velocity of energetic particles either. With the technique proposed here, the updating of the distribution function requires the solution of a linear system of equations, even though the BGK model is highly non linear. Thus the proposed schemes are particularly effective for high or moderate Mach numbers, where the macroscopic CFL condition is comparable to accuracy requirements. We show results for schemes of order 1 and 2, and the generalization to higher order is sketched.

S-ROCK: Chebyshev Methods for Stiff Stochastic Differential Equations

We present and analyze a new class of numerical methods for the solution of stiff stochastic differential equations (SDEs). These methods, called S-ROCK (for stochastic orthogonal Runge-Kutta Chebyshev), are explicit and of strong order 1 and possess large stability domains in the mean-square sense. For mean-square stable stiff SDEs, they are much more efficient than the standard explicit methods proposed so far for stochastic problems and give significant speed improvement. The explicitness of the S-ROCK methods allows one to handle large systems without linear algebra problems usually encountered with implicit methods. Numerical results and comparisons with existing methods are reported.

Krylov deferred correction accelerated method of lines transpose for parabolic problems

In this paper, a new class of numerical methods for the accurate and efficient solutions of parabolic partial differential equations is presented. Unlike traditional method of lines (MoL), the new Krylov deferred correction (KDC) accelerated method of lines transpose(MoLT) first discretizes the temporal direction using Gaussian type nodes and spectral integration, and symbolically applies low-order time marching schemes to form a preconditioned elliptic system, which is then solved iteratively using Newton–Krylov techniques such as Newton–GMRES or Newton–BiCGStab method. Each function evaluation in the Newton–Krylov method is simply one low-order time-stepping approximation of the error by solving a decoupled system using available fast elliptic equation solvers. Preliminary numerical experiments show that the KDC accelerated MoLT technique is unconditionally stable, can be spectrally accurate in both temporal and spatial directions, and allows optimal time-stepsizes in long-time simulations.

Implicit–Explicit Schemes for BGK Kinetic Equations

In this work a new class of numerical methods for the BGK model of kinetic equations is presented. In principle, schemes of any order of accuracy in both space and time can be constructed with this technique. The methods proposed are based on an explicit---implicit time discretization. In particular the convective terms are treated explicitly, while the source terms are implicit. In this fashion even problems with infinite stiffness can be integrated with relatively large time steps. The conservation properties of the schemes are investigated. Numerical results are shown for schemes of order 1, 2 and 5 in space, and up to third-order accurate in time.

General Linear Methods for Volterra Integro-differential Equations with Memory

A new class of numerical methods for Volterra integro-differential equations with memory is developed. The methods are based on the combination of general linear methods with compound quadrature rules. Sufficient conditions that guarantee global and asymptotic stability of the solution of the differential equation and its numerical approximation are established. Numerical examples illustrate the convergence and effectiveness of the numerical methods.

Implicit difference methods for parabolic functional differential equations

AbstractWe present a new class of numerical methods for the solution of quasilinear parabolic functional differential equations. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show by an example that the new methods are considerable better that the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators. Results obtained in the paper can be applied to differential integral problems and equations with retarded variables.

Implicit Difference Methods for First-Order Partial Differential Functional Equations

We present a new class of numerical methods for quasilinear first-order partial differential functional equations. The numerical methods are difference schemes implicit with respect to time variable. We give a complete convergence analysis for the methods and show by an example that the new methods are considerably better than explicit schemes. The proof of stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators with respect to the functional variable.

Computing Lyapunov exponents on a Stiefel manifold

The problem of numerical computation of a few Lyapunov exponents (LEs) of finite-dimensional dynamical systems is considered from the viewpoint of the differential geometry of Stiefel manifolds. Whether one computes one, many or all LEs of a continuous dynamical system by time integration, discrete or continuous orthonormalization is essential for stable numerical integration. A differential-geometric view of continuous orthogonalization suggests that one restricts the linearized vectorfield to a Stiefel manifold. However, the Stiefel manifold is not in general an attracting submanifold of the ambient Euclidean space: it is a constraint manifold with a weak numerical invariant. New numerical algorithms for this problem are then designed which use the fiber-bundle characterization of these manifolds. This framework leads to a new class of systems for continuous orthogonalization which have strong numerical invariance properties and the strong skew-symmetry property. Numerical integration of these new systems with geometric integrators leads to a new class of numerical methods for computing a few LEs which preserve orthonormality to machine accuracy. This idea is also taken a step further by making the Stiefel manifold an attracting invariant manifold in which case standard explicit Runge–Kutta algorithms can be used. This leads to an algorithm which requires only marginally more computation than a standard explicit integration without orthogonalization. These class of methods should be particularly effective for computing a few LEs for large-dimension dynamical systems. The new schemes are straightforward to implement. A test case is presented for illustration, and an example from dynamical systems is presented where a few LEs are computed for an array of coupled oscillators.

Element-Free Galerkin Simulations of Concrete Failure in Dynamic Uniaxial Tension Test

Dynamic tensile failure of concrete due to cracking is simulated using a new class of numerical methods, referred to as mesh-free methods, that are free of the traditional finite-element node-and-element data structure and are thus well suited to modeling the propagation of isolated tensile cracks. The results from a series of numerical simulations are compared with corresponding results from well-defined laboratory experiments involving the failure of concrete. Analysis of the numerical results, and their comparisons to the laboratory measurements, provide some insight into the subtleties in the experimental strain histories and improve understanding of the dynamic tensile failure of concrete.

Collocation-based two-step Runge-Kutta methods

In this paper we propose a new class of numerical methods of Runge-Kutta type for integrating initial-value problems for first-order ODEs, that is two-step Runge-Kutta (TRK) methods. The advantage of TRK methods is that for a given order of accuracyp we can construct methods of orderp with any desired number of implicit relations. In the very first investigation, the TRK methods are shown to be efficient numerical integration methods.

Primal-Dual Projected Gradient Algorithms for Extended Linear-Quadratic Programming

Many large-scale problems in dynamic and stochastic optimization can be modeled with extended linear-quadratic programming, which admits penalty terms and treats them through duality. In general, the objective functions in such problems are only piecewise smooth and must be minimized or maximized relative to polyhedral sets of high dimensionality. This paper proposes a new class of numerical methods for “fully quadratic” problems within this framework, which exhibit second-order nonsmoothness. These methods, combining the idea of finite-envelope representation with that of modified gradient projection, work with local structure in the primal and dual problems simultaneously, feeding information back and forth to trigger advantageous restarts.Versions resembling steepest descent methods and conjugate gradient methods are presented. When a positive threshold of $\varepsilon $-optimality is specified, both methods converge in a finite number of iterations. With threshold 0, it is shown under mild assumptions...

Projected Implicit Runge–Kutta Methods for Differential-Algebraic Equations

In this paper a new class of numerical methods, Projected Implicit Runge–Kutta methods, is introduced for the solution of index-2 Hessenberg systems of initial and boundary value differential-algebraic equations (DAEs). These types of systems arise in a variety of applications, including the modeling of singular optimal control problems and parameter estimation for differential-algebraic equations such as multibody systems. The new methods appear to be particularly promising for the solution of DAE boundary value problems, where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations. Previously defined symmetric methods have severe limitations when applied to these problems, including instability, oscillation, and loss of accuracy; the new methods overcome these difficulties. For linear problems we define an essential underlying boundary value ODE and prove well-conditioning of the differential (or state-space) solutio...

Numerical Methods for Constrained Equations of Motion in Mechanical System Dynamics

ABSTRACT ABSTRACT A new class of numerical methods for solving equations of motion of constrained mechanical systems is presented, the framework of which is based on manifold theoretic methods. Rewriting the system of differential-algebraic equations (DAEs) that describe constrained motion is ordinary differentia] equations (ODEs) on a constraint manifold, the theoretical framework for solving equations of motion is constructed, using a local