Method of lines solution of the Korteweg-de Vries equation

The Korteweg-de Vries equation (KdVE) is a classical nonlinear partial differential equation (PDE) originally formulated to model shallow water flow. In addition to the applications in hydrodynamics, the KdVE has been studied to elucidate interesting mathematical properties. In particular, the KDVE balances front sharpening and dispersion to produce solitons, i.e., traveling waves that do not change shape or speed. In this paper, we compute a solution of the KdVE by the method of lines (MOL) and compare this numerical solution with the analytical solution of the KdVE. In a second numerical solution, we demonstrate how solitons of the KdVE traveling at different velocities can merge and emerge. The numerical procedure described in the paper demonstrates the ease with which the MOL can be applied to the solution of PDEs using established numerical approximations implemented in library routines

A method of lines solution of the transient behavior of the helium cooled power leads for the SSC

In this study, a detailed numerical thermal mode of a 6.5 kA power lead for the Superconducting Super Collider has been developed, which was adapted from the dynamic model developed by Schiesser. The transient behavior of the power leads was modeled using, a method of lines (MOL) approach. The model was developed to pmvide a tool for analyzing coolant control strategies as well as an understanding of the behavior of the leads under presumed system transients. Results for a current ramp up to 4970 amps are favorably compared with measurements. Also, a loss of cooling situation is predicted to determine the transient temperature distribution under an off-design condition

A dynamic model for helium core heat exchangers

To meet the helium (He) requirements of the superconducting supercollider (SSC), the cryogenic plants must be able to respond to time-varying loads. Thus the design and simulation of the cryogenic plants requires dynamic models of their principal components, and in particular, the core heat exchangers. In this paper, we detail the derivation and computer implementation of a model for core heat exchangers consisting of three partial differential equations (PDES) for each fluid stream (the continuity, energy and momentum balances for the He), and one PDE for each parting sheet (the energy balance for the parting sheet metal); the PDEs have time and axial position along the exchanger as independent variables. The computer code can accommodate any number of fluid streams and parting sheets in an adiabatic group. Features of the code include: rigorous or approximate thermodynamic properties for He, upwind and downwind approximation of the PDE spatial derivatives, and sparse matrix time integration. The outputs from the code include the time-dependent axial profiles of the fluid He mass flux, density, pressure, temperature, internal energy and enthalpy. The code is written in transportable Fortran 77, and can therefore be executed on essentially any computer