On the stability of two new two-step explicit methods for the numerical integration of second order initial value problem on a variable mesh

Abstract

Initial value problems Variable mesh Two-step explicit method Damped wave equation Region of absolute stability Interval of periodicity Interval of weak stability Superstability a b s t r a c tWe present two novel two-step explicit methods for the numerical solution of the second order initial value problem on a variable mesh. In the case of a constant mesh the method is superstable in the sense of Chawla (1985). Numerical experimentation is provided to verify the stability analysis.

Spline-in-compression approximation of order of accuracy three (four) for second order non-linear IVPs on a graded mesh

“…The suggested estimation has been used to compute equations with undefined coefficients at the boundary point, and in particular, the last problem proves the success of the technique for such type of problem. Thus, discussed spline in compression estimation was revealed favourably in comparison with other estimations documented [ 6 , 15 , 17 ] in the literature.…”

“…In 2009, Saldanha and Saldanha [14] discussed a class of two-step explicit super-stable methods for a second-order linear IVP. An unconditionally stable technique for multi-dimensional time-dependent telegraphic equations using constant mesh was discussed in [ 15 ]. Mohanty and McKee [ 15 ] have studied the two-step explicit variable mesh method of order two for the second-order IVPs.…”

“…An unconditionally stable technique for multi-dimensional time-dependent telegraphic equations using constant mesh was discussed in [ 15 ]. Mohanty and McKee [ 15 ] have studied the two-step explicit variable mesh method of order two for the second-order IVPs. In 2017, Mohanty and Khurana [ 16 ] originally proposed spline-in-compression method for non-linear hyperbolic partial differential equations.…”

“…[ 20 ] developed a soliton-like solution on the dynamics of the transformed Peyrard-Bishop DNA model in the thermostat as a bio-fluid. It has been experienced in the past that the high accuracy numerical solution of IVPs fails to compute the solution near the vicinity of the singular point at t=0 [ 4 , 6 , 15 , 17 ]. We have overcome this difficulty by using three compact half-step points with the help of non-polynomial spline-in-compression approximation, which computes the solution in the entire solution domain for t > 0.…”

“…In order to validate the method, we solve the following examples using the procedure discussed in [ 6 , 15 , 17 ]. The explicit linear difference equations are computed directly, whereas nonlinear difference equations are computed using Newton method [22] , [23] , [24] .…”