About numerical solutions.

  The fact that the heat equation is an evolution equation simplifies how the solution to the heat conduction problem can be obtained numerically. We will postpone a complete discussion to a later time, but some issues in a typical computation are already self evident. For example, according to what we said above, to solve the heat conduction problem completely requires the determination of the temperature at all points in the strip in the x,t-plane figure 2. That would be an infinity of points and unknowns, something a digital computer cannot deal with. To make the number of unknowns finite, the points at which to find the temperature can be artificially restricted to a finite set by only including a mesh, or grid, of discrete points as shown in figure 4. (Of course, some approximations will need to be made to allow us to ignore the points that were ommitted, which could be done, for example, using a finite difference, nodal finite element, or spectral collocation procedure.)

  

Figure 4: Typical numerical solution of the heat equation.

Using a mesh, solving the heat conduction problem reduces finding the temperature values at a finite number of points. When a finer mesh, with more mesh points, is used, the accuracy will be higher, but so will the computational effort. Fortunately, for an evolution equation we do not have to try to find all these values at the same time. The values at the initial time plane in figure 4 are already known from the given initial condition (5). Now, if we look at the next higher time level, the temperature values can depend on the known initial values and the boundary conditions, but they cannot depend on the later time levels. As a result, we can find the temperature values at this second time level without worrying about the later ones. This makes the number of unknowns we have to deal with at once much smaller, which is usually a big advantage. After we have found the second time level, we can proceed with the third, and so on. In other words, we ``march'' the solution forward in time.