Abstract
We present a block hybrid functionally fitted Runge–Kutta–Nyström method (BHFNM) which is dependent on the stepsize and a fixed frequency. Since the method is implemented in a block-by-block fashion, the method does not require starting values and predictors inherent to other predictor-corrector methods. Upon deriving our method, stability is illustrated, and it is used to numerically solve the general second-order initial value problems as well as hyperbolic partial differential equations. In doing so, we demonstrate the method’s relative accuracy and efficiency.
Highlights
We consider the numerical solution of the second-order IVP y = f (x, y, y), y (x0) = y0, (1)y (x0) = y0, x ∈ [a, b], where f : R × Rm × Rm → Rm is a smooth function and m is the dimension of the system and its application to hyperbolic partial differential equations
We present a block hybrid functionally fitted Runge–Kutta–Nystrom method (BHFNM) which is dependent on the stepsize and a fixed frequency
Our objective is to present a block hybrid functionally fitted Runge–Kutta–Nystrom method (BHFNM) that is implemented in a block-by-block fashion, which does not suffer from the disadvantages of requiring starting values and predictors inherent to predictor-corrector methods
Summary
There are a wide range of methods that avoid orderreduction provided they have the special form y = f(x, y) These methods require less storage space and fewer function evaluations (cf Hairer [4], Hairer et al [7], Simos [8], Lambert et al, and [9], Twizell et al [10]). We propose a BHFNM which is of order 5 and its application is extended to solving oscillatory systems and hyperbolic partial differential equations. It is in the same spirit as those presented by Ngwane and Jator [30, 31].
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