Scalar Combustion Model

The moving mesh method solves well not only hyperbolic problems but also parabolic problems. The next example is a reaction-diffusion equation. It is described in Adjerid and Flaherty [1] as a model of a single step reaction with diffusion,

$$u_t=u_{xx}+D(1+a-u)\exp(-d/u),\quad 0< x<1, \quad 0<t,$$ (8)

$$u_x(0,t)=0, \quad u(1,t)=1, \quad 0<t,$$

$$u(x,0)=1, \quad 0\leq x\leq 1,$$

where D=Re^d/(ad) and R, d, a are constants. The solution represents the temperature of a reactant in a chemical system. For small times, the temperature gradually increases from unity with a ``hot spot'' forming at x=0. At a finite time ignition occurs, causing the temperature at x=0 to rapidly increase to 1+a. A front then forms and propagates towards x=1 with a very large speed (proportional to d). In most tests, a is close to 1, d=20 and R=5. A more difficult one is to set d=30 in (8). The wave front develops much faster than for d=20. So we choose the time scale $\tau=10^{-5}$, and the error tolerance RTOL and ATOL in DASSL to be 1.E-7. Indeed, for the Dorfi and Drury method, if $\tau \geq 10^{-4}$, the grid did not respond quickly to the changing of the solution, and the numerical results are not very good. The results of the moving mesh method with 40 nodes and results of fixed grid method with 200 and 500 nodes are shown in Fig. 4.

  

Figure: Dorfi and Drury (DD) method for scalar combustion model (d=30); N=41, $\tau =10^{-6}$. Output at t=0.0, 0.24, 0.2405, 0.241, 0.242, 0.244, 0.246.

We see that the moving mesh method with 40 points produces a better result than the fixed mesh method with 200 points, and is much more efficient than the fixed mesh method with 500 points, although they have almost the same accuracy. The comparison of performance is shown in Table 3.

 

Table 3: Computational performance of different methods for scalar combustion model (8),

Method	NSTP	NRES	NJAC	TNRES	EFN	CFN	CPU
fixed 500	5493	7851	35	8096	35	0	152.41
fixed 200	7228	11550	138	12516	104	0	85.90
moving 40	757	1408	78	2266	26	0	3.85

The local smoothing (LSM) method gets a similar result (see Fig. 5).

  

Figure: Local smoothing method (LSM) for scalar combustion model. N=41, $\tau =10^{-6}$.

However, LSM costs much more than DD with the same number of points. As in the Burgers' equation, the Dorfi and Drury method has a limitation on the minimum number of points. It cannot work for this problem (d=30) if the number of points is 21 or less. The LSM method has no such limitation. In Fig. 5, we see that the results with 21 points are acceptable.