A set of validated numerical integration methods based on explicit and implicit Runge-Kutta schemes is presented to solve, in a guaranteed way, initial value problems of ordinary differential equations. Runge-Kutta methods are well-known to have strong stability properties, which make them appealing to be the basis of validated numerical integration methods. A new approach to bound the local tr...

Exponential Runge-Kutta (ERK) and partitioned exponential Runge-Kutta (PERK) 4 methods are developed for solving initial value problems with vector fields that can be split into con5 servative and linear non-conservative parts. The focus is on linearly damped ordinary differential 6 equations, that possess certain invariants when the damping coefficient is zero, but, in the presence of 7 consta...

We study the stability of Runge-Kutta methods for the time integration of semidiscrete systems associated with time dependent PDEs. These semidiscrete systems amount to large systems of ODEs with the possibility that the matrices involved are far from being normal. The stability question of their Runge-Kutta methods, therefore, cannot be addressed by the familiar scalar arguments of eigenvalues...

in this paper a new isolating system is introduced for short to mid-rise buildings. in comparison to conventional systems such as lrb and hrb, the proposed system has the advantage of no need to cutting edge technology and has low manufacturing cost. this system is made up of two orthogonal pairs of pillow-shaped rollers that are located between flat bed and plates. by using this system in two ...

Runge-Kutta methods are an important family of implicit and explicit iterative methods used for the approximation of solutions of ordinary differential equations. Explicit RungeKutta methods are unsuitable for the solution of stiff equations as their region of stability is small. Stiff equation is a differential equation for which certain numerical methods for solving the equation are numerical...

8. First-Order Equations: Numerical Methods 8.1. Numerical Approximations 2 8.2. Explicit and Implicit Euler Methods 3 8.3. Explicit One-Step Methods Based on Taylor Approximation 4 8.3.1. Explicit Euler Method Revisited 4 8.3.2. Local and Global Errors 4 8.3.3. Higher-Order Taylor-Based Methods (not covered) 5 8.4. Explicit One-Step Methods Based on Quadrature 6 8.4.1. Explicit Euler Method Re...

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 analyze explicit Runge–Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first-order linear differential operator in space of Friedrichs-type. For the time discretization, we consider explicit secondand third-order Runge–Kutta schemes. We identify a general set of properties on the spatial stabilization, encompassing continuous a...

In this paper we analyze the consistency and stability properties of Runge-Kutta discrete adjoints. Discrete adjoints are very popular in optimization and control since they can be constructed automatically by reverse mode automatic differentiation. The consistency analysis uses the concept of elementary differentials and reveals that the discrete Runge-Kutta adjoint method has the same order o...