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Day02-slides - PD E O rder A tm s 502 CS 505 CSE 566 N u m...

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1 Atms 502 CS 505 CSE 566 Numerical Fluid Dynamics Tue., 29 August 2006 9/29/06 Atms 502 - Fall 2006 - Jewett 2 PDE Order The order of a PDE is that of the highest-order partial derivative We will typically look at problems that are first-order in time. "# " t = $ c "# " x " 2 u " x 2 + " 2 u " y 2 = 0 9/29/06 Atms 502 - Fall 2006 - Jewett 3 PDE linearity A linear PDE is linear in the unknown functions and their derivatives. … thus the coefficients multiplying each function or derivative depend only on independent variables. " u " t = # c " u " x " u " t = # u " u " x (linear) (nonlinear) 9/29/06 Atms 502 - Fall 2006 - Jewett 4 Nonlinear transition A numerical solution can also transition from a linear to a nonlinear regime . By this we mean a numerical solution might resemble a linearized theoretical solution of a physical problem. In such a case a small perturbation (disturbance) might grow exponentially - in a linear mode - until the perturbation is large enough that nonlinear processes take over.
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2 9/29/06 Atms 502 - Fall 2006 - Jewett 5 Nonlinear transition
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