Adaptive solution of non-linear fluid transmission lines

In fluid transmission line flow simulation problems it is advantageous to use a computational mesh that changes and adapts itself to suit the solution, using a dense mesh only in the neighbourhood of sharp changes, as in water hammer phenomena. Such adaptivity is difficult to achieve when the problem is solved using the conventional method of lines, where a fixed spatial discretization converts the partial differential equations (PDEs) into a system of coupled ordinary differential equation (ODE) initial value problem with time as the independent variable. An alternative is the transverse method of lines (also known as Rothe's method), where a temporal discretization of the PDEs yields a sequence of ODE boundary value problems (BVP) with position as the independent variable. Each BVP can be solved with its own mesh adapted to suit the shape of its solution. In this work we use the Rothe method to simulate the PDEs of one dimensional pipeline flow. Time discretization is done using a three-stage diagonally implicit Runge-Kutta method with constant time step. The BVPs are solved using the freely available Fortran code COLNEW, which uses a collocation method with adaptive mesh selection. The fluid flow equations incorporate, recently developed models of density and bulk modulus as functions of pressure.