Hyperbolic Equations

The most common situation yielding hyperbolic equations involves unsteady phenomena with convection. A prototype equation is

Depending on the interpretation of c and F(c) this can represent accumulation of mass and convection. With F(c) = u c, where u is the velocity, this equation represents a mass balance on concentration. If diffusive phenomenon are important, the equation is changed to

(1)

where D is a diffusion coefficient. As special cases we get the convective diffusive equation

(2)

and Burgers' equation with viscosity

(3)

where u is the velocity and n is the viscosity. This is a prototype equation for the Navier-Stokes equations in a shock. For adsorption phenomenon we might have [Rhee, et al., p. 202, 1986]

(4)

where f is the void fraction and f(c) gives the equilibrium relation between the concentration in the fluid phase and the concentration in the solid phase. In these examples, if the diffusion coefficient (D = 0) or viscosity (n = 0) are zero, the equations are hyperbolic. If D and n are small, the phenomenon may be essentially hyperbolic even though the equations are parabolic. Thus the numerical methods for hyperbolic equations may be useful even for parabolic equations.

Equations for several methods are given here, as taken from the book by Finlayson [1990]. If the convective term is treated with a centered difference expression the solution exhibits oscillations from node to node, and these only go away if a very fine grid is used. The simplest way to avoid the oscillations with a hyperbolic equation is to use upstream derivatives. If the flow is from left to right, this would give for equation (1)

and equation (3)

and equation (4)

If the flow were from right to left, then the formula would be

If the flow could be in either direction, a local determination must be made at each node i and the appropriate formula used. The effect of using upstream derivatives is to add artificial or numerical diffusion to the model. This can be ascertained by taking the finite difference form of the convective diffusion equation

and rearranging.

Thus the diffusion coefficient has been changed from

Another method often used for hyperbolic equations is the MacCormack method. This method has two steps, and it is written here for equation (2).

The concentration profile is steeper for the MacCormack method than for the upstream derivatives, but oscillations can still be present. The flux-corrected transport method can be added to the MacCormack method. A solution is obtained both with the upstream algorithm and the MacCormack method and then they are combined to add just enough diffusion to eliminate the oscillations without smoothing the solution too much. The algorithm is complicated and lengthy but well worth the effort. [Book, 1981; Sod, 1985; Finlayson, 1990].

If finite element methods are used, an explicit Taylor-Galerkin method is appropriate. For the convective diffusion equation the method is

Leaving out the Pe²t² terms gives the Galerkin method. Replacing the left-hand side with

gives the Taylor finite difference method, and dropping the Pe²t² terms in that gives the centered finite difference method. This method might require a small time-step if reaction phenomena are important. Then the implicit Galerkin method (without the Taylor terms) is appropriate.

The Taylor terms are not needed because the implicit time step provides the same effect as diffusion.

For the nonlinear equation (1) the Taylor-Galerkin method is

Figure. Stability Diagram for Convective Diffusion Equation. Stable below curve.

A stability diagram for the explicit methods applied to the convective diffusion equation is shown in the figure. Notice that all the methods require

where Co is the Courant number. How much Co should be less than one depends on the method and on r = Dt/x², as given in Figure 8.1. The MacCormack method with flux correction requires a smaller time step than the MacCormack method alone, and the implicit Galerkin method is stable for all values of Co and r shown in Figure 8.1 (as well as even larger values).

Each of these methods is trying to avoid oscillations which would disappear if the mesh were fine enough. For the steady convective diffusion equation these oscillations do not occur provided

(5)

For large Peclet number x must be small to meet this condition. An alternative is to use a small x in regions where the solution changes drastically. Since these regions change in time it is necessary that the elements or grid points move. The criteria to move the grid points can be quite complicated, and typical methods are reviewed in Finlayson [1990]. The criteria include moving the mesh in a known way (when the movement is known a priori), moving the mesh to keep some property (like first or second derivative measures) uniform over the domain, using a Galerkin or Weighted Residual criterion to move the mesh, and Euler-Lagrange methods which move part of the solution exactly by convection and then add on some diffusion after the fact.

The final illustration is for adsorption in a packed bed, or chromatography. Equation (4) can be solved when the adsorption phenomena is governed by a Langmuir isotherm.

Similar numerical considerations apply and similar methods are available [Finlayson 1990].