Introduction - Carnegie Mellon School of Computer Science

Introduction - Carnegie Mellon School of Computer Science

Partial Differential Equations Paul Heckbert Computer Science Department Carnegie Mellon University 7 Nov. 2000 15-859B - Introduction to Scientific Computing 1 Differential Equation Classes 1 dimension of unknown: ordinary differential equation (ODE) unknown is a function of one variable, e.g. y(t)

partial differential equation (PDE) unknown is a function of multiple variables, e.g. u(t,x,y) number of equations: single differential equation, e.g. y=y system of differential equations (coupled), e.g. y1=y2, y2=-g order nth order DE has nth derivative, and no higher, e.g. y=-g 7 Nov. 2000 15-859B - Introduction to Scientific Computing 2

Differential Equation Classes 2 linear & nonlinear: linear differential equation: all terms linear in unknown and its derivatives e.g. x+ax+bx+c=0 linear x=t2x linear x=1/x nonlinear 7 Nov. 2000 15-859B - Introduction to Scientific Computing 3

PDEs in Science & Engineering 1 Laplaces Equation: 2u = uxx +uyy +uzz = 0 unknown: u(x,y,z) gravitational / electrostatic potential Heat Equation: ut = a22u unknown: u(t,x,y,z) heat conduction Wave Equation: utt = a22u unknown: u(t,x,y,z) wave propagation 7 Nov. 2000 15-859B - Introduction to Scientific Computing

4 PDEs in Science & Engineering 2 Schrdinger Wave Equation quantum mechanics (electron probability densities) Navier-Stokes Equation fluid flow (fluid velocity & pressure) 7 Nov. 2000 15-859B - Introduction to Scientific Computing

5 2nd Order PDE Classification We classify conic curve ax2+bxy+cy2+dx+ey+f=0 as ellipse/parabola/hyperbola according to sign of discriminant b2-4ac. Similarly we classify 2nd order PDE auxx+buxy+cuyy+dux+euy+fu+g=0: b2-4ac < 0 elliptic (e.g. equilibrium) b2-4ac = 0 parabolic (e.g. diffusion) b2-4ac > 0 hyperbolic (e.g. wave motion) For general PDEs, class can change from point to point 7 Nov. 2000

15-859B - Introduction to Scientific Computing 6 Example: Wave Equation utt = c uxx for 0x1, t0 initial cond.: u(0,x)=sin(x)+x+2, ut(0,x)=4sin(2x) boundary cond.: u(t,0) = 2, u(t,1)=3 c=1 unknown: u(t,x) simulated using Eulers method in t k u discretize unknown function: j u (k t , j x )

7 Nov. 2000 15-859B - Introduction to Scientific Computing 7 Wave Equation: Numerical Solution 2 t u kj 1 2u kj u kj 1 c 2 u kj 1 2u kj u kj 1 x k+1 k

u0 = ... k-1 j-1 j j+1 u1 = ... for t = 2*dt:dt:endt u2(2:n) = 2*u1(2:n)-u0(2:n) +c*(dt/dx)^2*(u1(3:n+1)-2*u1(2:n)+u1(1:n-1)); u0 = u1; u1 = u2; end 7 Nov. 2000 15-859B - Introduction to Scientific Computing

8 Wave Equation Results dx=1/30 dt=.01 7 Nov. 2000 15-859B - Introduction to Scientific Computing 9 Wave Equation Results 7 Nov. 2000

15-859B - Introduction to Scientific Computing 10 Wave Equation Results 7 Nov. 2000 15-859B - Introduction to Scientific Computing 11 Poor results when dt too big dx=.05

dt=.06 Eulers method unstable when step too large 7 Nov. 2000 15-859B - Introduction to Scientific Computing 12 PDE Solution Methods Discretize in space, transform into system of IVPs Discretize in space and time, finite difference method.

Discretize in space and time, finite element method. Latter methods yield sparse systems. Sometimes the geometry and boundary conditions are simple (e.g. rectangular grid); Sometimes theyre not (need mesh of triangles). 7 Nov. 2000 15-859B - Introduction to Scientific Computing 13

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