The numerical solution of elliptic and parabolic partial differential equations with boundary singularities

Abstract

T h e p o t e n t i a l p r o b l e m t r e a t e d b y M o t z a n d W o o d s i s t a k e n a s a n u m e r i c a l e x a m p l e . T h e n u m e r i c a l r e s u l t s c o m p a r e f a v o u r a b l y w i t h t h o s e o b t a i n e d b y o t h e r t e c h n i q u e s .

Dynamics of a creeping Newtonian jet with gravity and surface tension: A finite difference technique for solving steady free‐surface flows using orthogonal curvilinear coordinates

“…If, however,. a more precise solution in the immediate vicinity of the singularity is desired, it may be fruitful to incorporate special numerical techniques (Crank and Furzeland, 1975) for treating boundary singularities into the basic computational scheme presented here. S T = surface tension parameter = total stress = components of total stress tensor = normal stress at the free surface = imposed normal stress at the free surface = tangential stress at the free surface = transverse velocity = axial velocity = components of velocity vector = average velocity = coordinate directions = coefficient in finite difference equation = axial distance…”

Modified symmetry technique for mitigation of flow leak near corners for compressible inviscid fluid flow

“…• J. Crank et al [6] listed various strategies employed in tackling corners for linear elliptic and parabolic equations,…”