An Almost L-Stable BDF-type Method
for the Numerical Solution of Stiff ODEs
Arising from the Method of Lines
Higinio Ramos, Jesús Vigo-Aguiar
Scientific Computing Group, Facultad de Ciencias, Universidad de Salamanca, Spain
Received 1 March 2006; accepted 10 March 2006
Published online 12 February 2007 in Wiley InterScience (www.interscience.wiley.com).
DOI 10.1002/num.20212
A new BDF-type scheme is proposed for the numerical integration of the system of ordinary differential
equations that arises in the Method of Lines solution of time-dependent partial differential equations. This
system is usually stiff, so it is desirable for the numerical method to solve it to have good properties concern-
ing stability. The method proposed in this article is almost L-stable and of algebraic order three. Numerical
experiments illustrate the performance of the new method on different stiff systems of ODEs after discretiz-
ing in the space variable some PDE problems. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential
Eq 23: 1110–1121, 2007
Keywords: BDF-type methods; stiff initial-value problems; A(α)-stability; almost L-stability; Method of
Lines
INTRODUCTION
In this article we consider a new numerical method for solving time-dependent partial differential
equations of the form
∂u
∂t
=
∂
∂x
r
(
x , t , u,
∂u
∂x
)
+ f
(
x , t , u,
∂u
∂x
)
,
(x , t) ∈[a, b]×[t
0
, t
f
],
(1)
with Dirichtlet boundary conditions given by
u(a, t) =
1
(t), u(b, t) =
2
(t), t ∈[t
0
, t
f
] (2)
and initial condition given as
u(x , t
0
) = φ(x), x ∈[a, b]. (3)
Correspondence to: Higinio Ramos, Scientific Computing Group, Facultad de Ciencias, Universidad de Salamanca,
Salamanca 37008, Spain (e-mail: higra@usal.es)
Contract grant sponsor: JCYL; contract grant number: SA 024/04
Contract grant sponsor: CYT; contract grant number: BMF 2004-295
© 2007 Wiley Periodicals, Inc.