Abstract
For the numerical integration of differential equations with oscillatory solutions an exponentially fitted explicit sixth-order hybrid method with four stages is presented. This method is implemented using variable step-size while its derivation is accomplished by imposing each stage of the formula to integrate exactly { 1 , t , t 2 , … , t k , exp ( ± μ t ) } where the frequency μ is imaginary. The local error that is employed in the step-size selection procedure is approximated using an exponentially fitted explicit fourth-order hybrid method. Numerical comparisons of the new and existing hybrid methods for the spring-mass and other oscillatory problems are tabulated and discussed. The results show that the variable step exponentially fitted explicit sixth-order hybrid method outperforms the existing hybrid methods with variable coefficients for solving several problems with oscillatory solutions.
Highlights
Computation of the solutions of the special second order initial value problems y00 (t) = f (t, y(t)), y(t0 ) = y0, y0 (t0 ) = y0 0 in which the first derivative does not appear explicitly has spawned many numerical algorithms and approaches
For variable step-size implementation where the step-size varies throughout the integration, accuracy and the computational cost are very much depending on the formulation of the numerical methods and the step-size control algorithm
EXH6: The exponentially fitted explicit sixth-order hybrid method with four stages derived in this paper
Summary
It is to observed that more attention is paid to properties than the interval in the than development of embedded formulas for variable the algebraic orders andof absolute phase-lagstability properties the interval of absolute stability in the step-size implementation. For hybrid methods corresponding to Equation (4), the interval (0, Ha ) is called the interval of absolute stability if P(H2 ) < 1 and S(H2 ) < 1 + P(H2 )for all H ∈ (0, Ha ). If the coefficients of hybrid methods defined in Equation (1) are functions of v = μh where μ is the frequency of the problem and h is the step-size, the interval of absolute stability is generalized to the region of stability.
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