Notes u No extra class tomorrow cs 542

Notes u No extra class tomorrow cs 542 g-term 1 -2007 1

PDE’s u Subject of Partial Differential Equations is vast u We’ll focus on one particularly important equation: • Called Poisson’s equation (if right hand side is zero, Laplace’s equation) u Arises • • • almost everywhere Minimization of norm of gradient (see RBF’s) Gravitational/electrostatic potential Steady state of heat flow and other diffusion processes Stochastic processes (Brownian motion) Fluid dynamics cs 542 g-term 1 -2007 2

Reduce to 1 D first u Typical Boundary Value Problem (BVP): u Boundary conditions: specify solution value: “Dirichlet” specify solution derivative: “Neumann” u Can’t directly solve as a time integration cs 542 g-term 1 -2007 3

Finite Difference Method u Discretize unknown solution on a grid: u Use Taylor series to estimate derivatives from values on grid cs 542 g-term 1 -2007 4

Discretized Boundary Conditions u Dirichlet: substitute in known values u Neumann: discretize boundary condition, use it to extrapolate cs 542 g-term 1 -2007 5

Solve! u At each grid point we have a linear equation u Combine into one large linear system (solve for all solution values simultaneously) u Resulting matrix is symmetric (negative) definite, and sparse • In fact, in 1 D, just tridiagonal… cs 542 g-term 1 -2007 6

Higher dimensions u Lay down a regular grid as before u Matrices get even bigger, but not quite as simple structure u Notion of stencil: shorthand for matrix cs 542 g-term 1 -2007 7

Stability u The preceding methods work, but not every stencil does u Need a notion of stability ~ conditioning u Example problem with central differences • Matrix could be singular, or worse u Example problem with one-sided differences • Information propagation is wrong cs 542 g-term 1 -2007 8

Finite Difference Limitations u Accurately treating boundary conditions that don’t line up with the grid u Adaptivity cs 542 g-term 1 -2007 9