Comparison of method of lines and finite difference solutions of 2‐D Navier–Stokes equations for transient laminar pipe flow

Abstract

SUMMARYPerformances of method of lines (MOL) and ÿnite di erence method (FDM) were tested from the viewpoints of solution accuracy and central processing unit (CPU) time by applying them to the solution of time-dependent 2-D Navier-Stokes equations for transient laminar ow without=with sudden expansion and comparing their results with steady-state numerical predictions and measurements previously reported in the literature. Predictions of both methods were obtained on the same computer by using the same order of spatial discretization and the same uniform grid distribution. Axial velocity and pressure distribution in pipe ow and steady-state reattachment lengths in sudden expansion ow on uniform grid distribution predicted by both methods were found to be in excellent agreement. Transient solutions of both methods for pipe ow problem show favourable comparison and are in accordance with expected trends. However, non-physical oscillations were produced by both methods in the transient solution of sudden expansion pipe ow. MOL was demonstrated to yield non-oscillatory solutions for recirculating ows when intelligent higher-order discretization scheme is utilized for convective terms. MOL was found to be superior to FDM with respect to CPU and set-up times and its exibility for incorporation of other conservation equations.

Leveraging PDE solver for predicting transient space-use dynamics in ecological and epidemiological systems

“…The procedure works by first discretizing only the spatial components of a PDE, thereby approximating it with a system of ordinary differential equations which are subsequently solved under discretized time. Although both approaches produce comparatively accurate solutions for 2-dimensional advection-diffusion problems (Selçuk et al 2002), MOL’s inability to handle elliptic PDEs, i.e., when the time-derivative is removed, can preclude one from evaluating aspects of the transient dynamics with respect to the steady-state solution unless the latter is already known in its analytical form (but see R package ReacTran).…”

The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition

“…They showed (experimentally) that increasing the points in the formula in the spatial approximation cannot bring the MOL and exact solutions into close agreement(for parabolic problems but not necessarily for hyperbolic or elliptic equations). The method of lines and finite difference method were tested in [43] from the viewpoints of solution accuracy and central processing unit time by applying them to the solution of time-dependent two-dimensional Navier-Stokes equations for transient laminar flow without/with sudden expansion and comparing their results with steady-state numerical predictions and measurements previously reported in the literature. MOL was found to be superior to finite difference method with respect to CPU and set-up times and its flexibility for incorporation of other conservative equations [43].…”

“…The method of lines and finite difference method were tested in [43] from the viewpoints of solution accuracy and central processing unit time by applying them to the solution of time-dependent two-dimensional Navier-Stokes equations for transient laminar flow without/with sudden expansion and comparing their results with steady-state numerical predictions and measurements previously reported in the literature. MOL was found to be superior to finite difference method with respect to CPU and set-up times and its flexibility for incorporation of other conservative equations [43]. The method of lines concept has been combined with the boundary element based elimination of the spatial derivatives to obtain a solution method for partial differential equations which are parabolic in time [44].…”

Mol solution for transient turbulent flow in a heated pipe

“…The former can be achieved by increasing the order of spatial discretization method, resulting in high accuracy with less grid points, and using not only highly accurate but also a stable numerical algorithm for time integration. The method of lines, the superiority of which over finite difference method had already been proven [1], is an alternative approach that meets this requirement for the time dependent problems. The latter requirement is met by either supercomputers or parallel computers which require efficient parallel algorithms.…”