The second-order delay differential equations often appear in the dynamical system, celestial mechanics, kinematics, and so forth. Some numerical methods for solving secondorder delay differential equations have been discussed, which include θ-method [1], trapezoidal method [2], and RungeKutta-Nyströmmethod [3].The variational iteration method (VIM) was first proposed by He [4, 5] and has been ...

We study the convergence properties of implicit Runge-Kutta methods applied to time discretization of parabolic equations with timeor solutiondependent operator. Error bounds are derived in the energy norm. The convergence analysis uses two different approaches. The first, technically simpler approach relies on energy estimates and requires algebraic stability of the RungeKutta method. The seco...

Submitted for the DFD11 Meeting of The American Physical Society A coarse-grid projection method for accelerating incompressible flow computations OMER SAN, ANNE STAPLES, Virginia Tech — We present a coarse-grid projection (CGP) algorithm for accelerating incompressible flow computations, which is applicable to methods involving Poisson equations as incompressibility constraints. CGP methodolog...

Ordinary differential equations (ODEs) are often used to model the dynamics of (often safety-critical) continuous systems. This work presents the formal verification of an algorithm for reachability analysis in continuous systems. The algorithm features adaptive RungeKutta methods and rigorous numerics based on affine arithmetic. It is proved to be sound with respect to the existing formalizati...

High-order semi-implicit Picard integral deferred correction (SIPIDC) methods have previously been proposed for the time-integration of partial differential equations with two or more disparate time scales. The SIPIDC methods studied to date compute a high-order approximation by first computing a provisional solution with a first-order semi-implicit method and then using a similar semiimplicit ...

A major problem in obtaining an efficient implementation of fully implicit RungeKutta (IRK) methods applied to systems of differential equations is to solve the underlying systems of nonlinear equations. Their solution is usually obtained by application of modified Newton iterations with an approximate Jacobian matrix. The systems of linear equations of the modified Newton method can actually b...

A major problem in obtaining an efficient implementation of fully implicit RungeKutta (IRK) methods applied to systems of differential equations is to solve the underlying systems of nonlinear equations. Their solution is usually obtained by application of modified Newton iterations with an approximate Jacobian matrix. The systems of linear equations of the modified Newton method can actually b...

Abstract: This paper addresses the numerical solution of optimal control problems for systems described by ordinary differential equations with control constraints. The state equation is discretized by a general explicit Runge-Kutta scheme and the controls are approximated by functions that are piecewise polynomial, but not necessarily continuous. We then propose an approximate gradient project...

Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source terms. In our earlier work [31–33], we designed high order well-balanced schemes to a class of hyperbolic systems with separable source terms. In this paper, we present a different approach to the same purpose: designing high order well-balanced finite volume weight...

In this paper, we present an efficient semi-Lagrangian based particle level set method for the accurate capturing of interfaces. This method retains the robust topological properties of the level set method without the adverse effects of numerical dissipation. Both the level set method and the particle level set method typically use high order accurate numerical discretizations in time and spac...