We study the linear stability of partitioned Runge–Kutta (PRK) methods applied to linear separable Hamiltonian ODEs and to the semidiscretization of certain Hamiltonian PDEs. We extend the] by presenting simplified expressions of the trace of the stability matrix, tr Ms, for the Lobatto IIIA–IIIB family of symplectic PRK methods. By making the connection to Padé approximants and continued fract...

This study computes the gradient of a function numerical solutions ordinary differential equations (ODEs) with respect to initial condition. The adjoint method approximately by solving corresponding system numerically. In this context, Sanz-Serna [SIAM Rev., 58 (2016), pp. 3–33] showed that when value problem is solved Runge–Kutta (RK) method, can be exactly computed applying an appropriate RK ...

We study the spatial semidiscretizations obtained by applying Runge–Kutta (RK) and partitioned Runge–Kutta (PRK) methods to multisymplectic Hamiltonian partial differential equations. These methods can be regarded as multisymplectic hp-finite element methods for wave equations. All the methods we consider are multisymplectic; we determine their properties with regard to existence of solutions, ...

We derive order conditions for the discretization of (unconstrained) optimal control problems, when the scheme for the state equation is of Runge-Kutta type. This problem appears to be essentially the one of checking order conditions for symplectic partitioned Runge-Kutta schemes. We show that the computations using bi-coloured trees are naturally expressed in this case in terms of oriented fre...

One of the problems in computational aeroacoustics (CAA) is the large disparity between the length and time scales of the flow field, which may be the source of aerodynamically generated noise, and the ones of the resulting acoustic field. This is the main reason why numerical schemes, used to calculate the timeand space-derivatives, should exhibit a low dispersion and dissipation error. This p...

We consider the preservation of weak solution invariants in the time integration of ordinary diier-ential equations (ODEs). Recent research has concentrated on obtaining symplectic discretizations of Hamiltonian systems and schemes that preserve certain rst integrals (i.e. strong invariants). In this article, we examine the connection between constrained systems and ODEs with weak invariants fo...

In this article we consider partitioned Runge-Kutta (PRK) methods for Hamiltonian partial differential equations (PDEs) and present some sufficient conditions for multi-symplecticity of PRK methods of Hamiltonian PDEs.

This paper is concerned with time-stepping numerical methods for computing stiff semi-discrete systems of ordinary differential equations for transient hypersonic flows with thermo-chemical nonequilibrium. The stiffness of the equations is mainly caused by the viscous flux terms across the boundary layers and by the source terms modeling finite-rate thermo-chemical processes. Implicit methods a...