Abstract
The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method.
Highlights
We will consider a suitable modification of multi-derivative onestep methods derived in [1] for the numerical solution of first order differential equations y (t) = f (y(t)), t ∈ [t0, T], (1)
In comparison with the original multi-derivative midpoint (MDMP) method we see that the computational cost for the implementation of the new formulae decreases, because we just need to add the computation of the two explicit steps in the nonlinear iteration
The method (12) has the special property of being defined as the composition of two formulae: this suggests that the method obtained by the reverse composition is conjugate to (12), and we get a couple of symplectic/conjugatesymplectic Runge–Kutta schemes that extend to the fourth-order the well-known couple formed by the midpoint and trapezoidal methods
Summary
Since a certain degree of freedom is allowed in the choice of the derivative discretization stepsize, it turns out that the final full discretized formula may exhibit more favorable geometric properties with respect to the original one This is the case for two special methods in the classes we are going to introduce: they form a pair of symplectic and conjugate-symplectic Runge–Kutta integrators that originate from the midpoint method and its conjugate-symplectic counterpart, namely the trapezoidal methods. We introduce and analyze a new technique for solving the nonlinear systems emerging from the implementation of the methods It consists of a block-diagonal variant of the simplified Newton scheme which requires, at each integration step, a single Jacobian evaluation of the vector field and a single LU factorization of a matrix having the same size of the continuous problem.
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