Page 190 - Numerical methods for chemical engineering

176 4 Initial value problems

Overview of ODE-IVP solvers in MATLAB

We next provide an overview of ODE-IVP solvers. The contents of this section provide a
sufficient background to solve problems of the form

˙ x = f (x; Θ) x(t 0 ) = x [0] (4.113)
First, we describe the basic time-marching approach of ODE-IVP solvers and contrast
explicit and implicit, single-step and multistep solvers. Then, we demonstrate the use of the
explicit single-step solver ode45 and the implicit multistep solver ode15s.

Time-marching ODE-IVP solvers

ODE solvers update x(t) in discrete time steps of size t to compute x(t k ) at times t 0 < t 1
< t 2 < ··· . For a constant time step, t k = t 0 + k( t); but often t varies throughout the
course of the simulation. t is smaller for greater accuracy when x(t) changes rapidly and
is larger for increased simulation speed when x(t) changes slowly. Over each time step, the
exact update of the state vector is
' t k + t ' t k + t
x(t k + t) − x(t k ) = ˙ x(t)dt = f (x(t); Θ)dt (4.114)
t k t k
If x [k] = x(t k ), the new state at t k+1 = t k + t can be estimated by a rule that approximates
(4.114), of the form
[k+1] [k] [k] [k+1] [k+1]
x − x = ( t)F x , x , f (x; Θ) x ≈ x(t k+1 ) (4.115)
[k]
F[x , x [k+1] , f (x; Θ)] is a rule involving the old and new states that defines the method.
At each step, there is introduced a new local error, proportional to some power of t, that
is the difference between the approximate update (4.115) and the exact update (4.114).
[k]
As (4.115) uses information only about the state values at the beginning, x , and end,
x [k+1] , of the current time step, it is said to define a single-step integration method.For
example, in the Crank–Nicholson method

[k] [k+1] 1 [k] [k+1]
F CN x , x , f (x; Θ) ≡ f x ; Θ + f x ; Θ (4.116)
2
and in the implicit (backward) Euler method
[k] [k+1] [k+1]
F BE x , x , f (x; Θ) ≡ f x ; Θ (4.117)
Because f (x; Θ) generally is nonlinear, (4.115) often cannot be rearranged to provide a
direct expression for x [k+1] . Then, (4.115) is said to generate an implicit integration method
that requires a nonlinear algebraic system to be solved at each time step.

Explicit single-step methods

By contrast, an explicit single-step integration method
[k]

x [k+1] − x [k] = ( t)F x , f (x; Θ) (4.118)