Page 286 - Numerical Methods for Chemical Engineering

Finite differences for a convection/diffusion equation 275

dimensions, it is common to add numerical diffusion only in the streamline direction, to
obtain an effective anisotropic diffusion tensor,
vv T
= I + num 2 (6.92)
|v|
For an in-depth treatment of methods for BVPs with strong convection, see Finlayson
(1992).

Numerical solution of the HJB equation of optimal control

We now are in a position to analyze the numerical solution of the HJB equation from optimal
control (Chapter 5) to minimize the cost functional
'
t H
[0]
F u(t); x = σ(s, x(s), u(s))ds + π(x(t H )) (6.93)
t 0
for a system governed by ˙x = f (t, x, u). In terms of the backward time τ = t H − t, we
solve the HJB equation
∂ϕ
= min u(τ,x) [σ(t H − τ, x, u) + ∇ϕ · f (t H − τ, x, u)] (6.94)
∂τ
with the initial condition ϕ(0, x) = π(x). The optimal value of the cost functional
[0]

F[u(t); x ]is ϕ τ = t H − t 0 , x [0] and the “best” feedback control law for the system
is u con (x) = u(τ = t H − t 0 , x). In Chapter 5 we solved the HJB equation for a simple 1-D
control problem, and are now in a position to explain the choice of discretization used
there. We now recognize that (6.94) is a reaction/convection equation with σ taking the
role of a source term and f taking the role of −v. Thus, at each point in the calcula-
tion, we should examine the local sign of f and choose the appropriate upwind one-sided
difference. This is in fact what was done in the example of Chapter 5, but without expla-
2
nation. In practice, it is also common to add an artificial diffusion term ε∇ ϕ to the HJB
equation to obtain a “viscosity solution.” Diffusion makes dynamic programming more
robust as it smooths out any discontinuities in the dynamics, cost functional, or control
input.
When solving (6.94), the spatial domain should be extended to very large positive and
negative values of x to limit any corruption of the solution in the region of interest by the
boundary condition employed at the limits. To see how one might treat the boundaries, refer
again to the example in Chapter 5.

Characteristics and types of PDEs

We see from our discussion of the 1-D convection/diffusion equation that the qualitative
nature of an equation, and of its numerical solution, can change as we vary the parameters.
These changes are related to the nature of how information about the field is propagated by
the differential equation in space and time. The importance of information flow is reflected
in a common naming convention for second-order PDEs. As this nomenclature is employed
often in the literature, we briefly review it here.