High accuracy A.D.I. method s for fourth order parabolic
equations with variable coefficients
Celia Andrade
(*)
and S. McKee
(**)
ABSTRACT
High accuracy alternating direction implicit (A. D.I.) methods
are
derived for solving fourth order
parabolic equations with variable coefficients in one, two, and three space dimensions. Splittings
are discussed and numerical results are presented.
1.
INTRODUCTION 2. TWO SPACE DIMENSIONS
Consider the linear parabolic equation
a2u + a(x,t) a4u =0, a(x,t)> 0
at 2 ax 4
(1.1)
in the region of the (x, t) plane given by I x (0 g t • T)
where I -- (0 • x g 1) and x, t are the distance and
time co-ordinates respectively. If the region is covered
by a rectilinear grid with h, k the grid spacings in the
x- and t-directions respectively, we can show by Taylor
expansion of the operators that the difference equa-
tion
r 2 n+l : un+l
i a n 62(a )-1+__ a ~ ]
[I+--~-
o
x 12
P o
~)-1 r 2 n 4 U n
1 a n 62(a +~a 6 I
-211+--~ o x 12 o x 0
+11+-~-1 a no °x'2"~ao)n'-l+ r212 an-lo 8:]U n-lo
=r2a ,x 4
# (1.2)
is locally accurate to terms that are 0 (h 4) = 0 (k 2) at
the grid points x= oh, t=nk (1 ¢ o ¢ M-l, 1 ¢ n¢ N-l)
where Mh= 1, Nk=T. In (1.2) U~ is the solution of
the difference equation at the grid point x = oh, t = nk,
a n is the value of a at x = oh, t = nk, r is the mesh ratio
f/h2
and 8 2 is the usual central difference operator in
the x-direction. It is assumed that u e C 6 and a e" C 4.
It is the purpose of this note to extend this scheme to
two space dimensions, in which case the linear para-
bolic equation under consideration will be
a2U+a(x,y,t)~a4u +b(x,y,t)--a4u=0
at 2 ax 4 ay 4
a (x, ys t) > 0, b (x, y, t) > 0 (1.3)
It is convenient to consider the initial boundary value
problem consisting of (1.3) together with
u (x, y, o) = fl (x, y)
~ (x, y, o) = f2 (x, y) jt- (x,y) e.
u (Xs y, t) = gl (x, y, t)
82u (x, y, t) = g2 (x, y, t)
~x 2 (x,y)e a f~, 0~ t,; T
a2u
-- (x, y, t) = g3 (x, y, t)
ay 2
where f~ is a connected region in the (x, y) plane and
an is its boundary. The region ~ x (0,; t,; T) is
covered by a rectilinear grid with grid spacings h, k
in the space and time co-ordlnates respectively. If f~
is a unit square, the internal grid points are given by
x=ah, y=mh (lgp, mg M-1)whereMh=l.
For the particular case a(x, y, t) = b (x, y t) = 1 it is
simple to derive the high accuracy formula
(. n+l
1 2 1 2.4,(1+ 1..1._62 + 1 r28:) Op, m
(1+-6-6x+'~ r 6x) 6 Y
12
n-1
-
2U~, m +
Up, m)
=-r2[s:+84+ 1 264+6284)IU n
y "-~-(Sx y y x p,m (2.1)
where U n is the value of U at the grid point x = oh,
Ps m
y=mhs t=nk.
By performing a similar analysis to that in McKee (1973)
and Gourhy and Mitchell (1968) we obtain the dif-
ference scheme
(*) Celia Andrade, Instituto de Ci~ncias Matem~ticas de S~:o Carlos, S~o Carlos, S~o
Paulo.
(**) S. McKee, Hertford College and The Computing Laboratory, Oxford.
Journal of Computational and Applied Mathematics, volume 3, no 1, 1977. 11
