Implicit Runge-Kutta(IRK) methods for solving the nonsmooth ordinary differential equation (ODE) involve a system of nonsmooth equations. We show superlinear convergence of the slanting Newton method for solving the system of nonsmooth equations. We prove the slanting differentiability and give a slanting function for the involved function. We develop a new code based on the slanting Newton met...

Recently, a remarkable correspondence has been unveiled between a certain class of ordinary linear differential equations (ODE) and integrable models. In the first part of the report, we survey the results concerning the 2nd order differential equations , the Schrödinger equation with a polynomial potential. We will observe that fundamental objects in the study of the solvable models, e.g., Bax...

Recently, a remarkable correspondence has been unveiled between a certain class of ordinary linear differential equations (ODE) and integrable models. In the first part of the report, we survey the results concerning the 2nd order differential equations , the Schrödinger equation with a polynomial potential. We will observe that fundamental objects in the study of the solvable models, e.g., Bax...

A cell-centered implicit-explicit updated Lagrangian finite volume scheme on unstructured grids is proposed for a unified first order hyperbolic formulation of continuum fluid and solid mechanics. The provably respects the stiff relaxation limits continuous model at fully discrete level, thus it asymptotic preserving. Furthermore, GCL satisfied by compatible discretization that makes use nodal ...

Runge-Kutta (RK) methods may exhibit order reduction when applied to certain stiff problems. While fully implicit RK schemes exist that avoid via high-stage order, DIRK (diagonally Runge-Kutta) are practically important due their structural simplicity; however, these cannot possess high stage order. The concept of weak (WSO) can also overcome reduction, and it is compatible with the structure. ...

This paper introduces a new method for solving ordinary differential equations (ODEs) that enhances existing methods that are primarily based on finding integrating factors and/or point symmetries. The starting point of the new method is to find a non-invertible mapping that maps a given ODE to a related higher-order ODE that has an easily obtained integrating factor. As a consequence, the rela...

The solution space of a constant coefficient ODE gives rise to a natural real analytic curve in Euclidean space. We give necessary and sufficient conditions on the ODE to ensure that this curve is a proper embedding of infinite length or has finite total first curvature. If all the roots of the associated characteristic polynomial are simple, we give a uniform upper bound for the total first cu...