Numerical Methods for Partial Differential Equations (16.920J/2097J/SMA5212)
Course Outline ● Overview of pdes(1) o Finite differences methods(6) Finite volume methods(3) Finite element methods(7) Boundary integral methods(6) Solution methods (3) Total: 26 lectures
Assessment Four Problem Sets/Mini-projects Finite Differences 25% Hyperbolic Equations 20% Finite Elements 25 Boundary Integral Methods 20% Class Interaction 10%
Partial Differential Equations An Overview Lecture 1
Convection-Diffusion Model Equation al ot+U Vu=kVu+f N1 66 =(8x0y da U, k>0, f, given functions of(ec, y) Scalar, Linear, Parabolic equation N2 SMA-HPC⊙2003MT Partial Differential Equations 1
Convection-Diffusion Model Equation Applications du o+U·V=kV2+f If a is o Temperature> Heat Transfer e Pollutant Concentration>Coastal Engineering o Probability Distribution,Statistical Mechanics Price of an Option Financial Engineering SMA-HPC⊙2003MT Partial Differential Equations 2
Elliptic Equations Limiting Cases Poisson Equation KVu=f in Convection-Diffusion U. Vu=kVu in Q . Smooth" solutions The domain of dependence of u(e, y) is 2 SMA-HPC⊙2003MT Partial Differential Equations 3
Parabolic Equations Limiting Cases Heat equation 8u_Kv20+f in s t " Smooth” solutions The domain of dependence ot u(a,y, T)is(a, g, t<T) SMA-HPC⊙2003MT Partial Differential Equations 4
Hyperbolic Equations Limiting Cases Wave Equation(First order du Ot+U. Vu=f in n2 Non-smooth solutions o Characteristics: d=U(ec(t)) The domain of dependence of u(a, T)is(ec(t),t< T SMA-HPC⊙2003MT Partial Differential Equations 5
Hyperbolic Equations Limiting Cases Convection Equation U·Vu=fins T Non-smooth solutions o Characteristics are streamlines of U,e.g. ds=U The domain of dependence of u(ac)is(acs),s<0) SMA-HPC⊙2003MT Partial Differential Equations 6
