An accelerated splitting-up method for parabolic equations
Abstract
We approximate the solution
u
of the Cauchy problem
\frac{\partial}{\partial t} u(t,x)=Lu(t,x)+f(t,x), \quad
(t,x)\in(0,T]\times\bR^d,
u(0,x)=u_0(x),\quad x\in\bR^d
by splitting the equation into the system
∂
∂t
v
r
(t,x)=
L
r
v
r
(t,x)+
f
r
(t,x),r=1,2,...,
d
1
,
where
L,
L
r
are second order differential operators,
f
,
f
r
are functions of
t,x
, such that
L=
∑
r
L
r
,
f=
∑
r
f
r
. Under natural conditions on solvability in the Sobolev spaces
W
m
p
, we show that for any
k>1
one can approximate the solution
u
with an error of order
δ
k
, by an appropriate combination of the solutions
v
r
along a sequence of time discretization, where
δ
is proportional to the step size of the grid. This result is obtained by using the time change introduced in [7], together with Richardson's method and a power series expansion of the error of splitting-up approximations in terms of
δ
.