The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
A new basis implementation for a mixed order boundary value ODE solver
SIAM Journal on Scientific and Statistical Computing
SIAM Journal on Scientific and Statistical Computing
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Defect correction for two-point boundary value problems on nonequidistant meshes
Mathematics of Computation
Runge-Kutta Software with Defect Control four Boundary Value ODEs
SIAM Journal on Scientific Computing
Runge-Kutta methods for the solution of stiff two-point boundary value problems
Applied Numerical Mathematics - Special issue celebrating the centenary of Runge-Kutta methods
On the generation of mono-implicit Runge-Kutta-Nystro¨m methods by mono-implicit Runge-Kutta methods
Journal of Computational and Applied Mathematics
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Mono-implicit Runge-Kutta (MIRK) formulae are widely used for the numerical solution of first order systems of nonlinear two-point boundary value problems. In order to avoid costly matrix multiplications, MIRK formulae are usually implemented in a deferred correction framework and this is the basis of the well known boundary value code TWPBVP. However, many two-point boundary value problems occur naturally as second (or higher) order equations or systems and for such problems there are significant savings in computational effort to be made if the MIRK methods are tailored for these higher order forms. In this paper, we describe MIRK algorithms for second order equations and report numerical results that illustrate the substantial savings that are possible particularly for second order systems of equations where the first derivative is absent.