1. Parabolic PDEs Generally involve change of quantity in space and time Equivalent to our previous example - heat conduction

2. Can use the same grid idea, only boundaries are different time (j index) space (i index)

3. Need initial conditions - green dots and boundary conditions - red dots One side of grid is always open

4. Because parabolic PDEs are open-ended in time, can have instability Need to address this in solution methods Explicit Implicit

5. Explicit solutions Use finite differences x i-1 xixi x i+1 tjtj t j+1

6. Substitute into equation and rearrange Explicit solution for this parabolic PDE

7. Example: Given the initial condition and boundary conditions below, solve for heat distribution over time See matlab code

8. Stability conditions It can be shown that the condition for stability is or look at what happens when you play with lambda

9. Other points Can include derivative boundary conditions - introduce an imaginary point as in ellipitical example Use finite difference for derivative to eliminate imaginary point, introduce derivative into propagation equation

10. We get

11. Can also use higher order temporal approximations for time term (and for space term)

12. Implicit methods avoid stability problems One example implicit method - use next time step to approximate spatial derivative

13. Substituting, Three unknowns

14. Write these equations for all interior nodes - you get enough equations Set up in matrix and get a tridiagonal matrix Unconditionally stable - although accuracy is degraded at larger  t