ﻻ يوجد ملخص باللغة العربية
Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were proposed and analyzed in [8]. These specially designed methods use reduced precision or the implicit computations and full precision for the explicit computations. We develop a FORTRAN code to solve a nonlinear system of ordinary differential equations using the mixed precision additive Runge-Kutta (MP-ARK) methods on IBM POWER9 and Intel x86_64 chips. The convergence, accuracy, runtime, and energy consumption of these methods is explored. We show that these MP-ARK methods efficiently produce accurate solutions with significant reductions in runtime (and by extension energy consumption).
In this work we consider a mixed precision approach to accelerate the implemetation of multi-stage methods. We show that Runge-Kutta methods can be designed so that certain costly intermediate computations can be performed as a lower-precision comput
This study computes the gradient of a function of numerical solutions of ordinary differential equations (ODEs) with respect to the initial condition. The adjoint method computes the gradient approximately by solving the corresponding adjoint system
We categorify the RK family of numerical integration methods (explicit and implicit). Namely we prove that if a pair of ODEs are related by an affine map then the corresponding discrete time dynamical systems are also related by the map. We show that
It is well-known that a numerical method which is at the same time geometric structure-preserving and physical property-preserving cannot exist in general for Hamiltonian partial differential equations. In this paper, we present a novel class of para
In this paper, a family of arbitrarily high-order structure-preserving exponential Runge-Kutta methods are developed for the nonlinear Schrodinger equation by combining the scalar auxiliary variable approach with the exponential Runge-Kutta method. B