Introduction to Numerical Methods for ODEs and PDEs Lectures 1 and 2: Some representative problems from engineering science and classification of equation.

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Presentation transcript:

Introduction to Numerical Methods for ODEs and PDEs Lectures 1 and 2: Some representative problems from engineering science and classification of equation types

Strategy for the next four lectures Goal: to survey important considerations in the application of numerical methods to ordinary and partial differential equations in engineering and the sciences Topics: Introduction to common types (with examples) of boundary value problems Finite difference approximations –Integration of ODEs in time –Approximation of PDEs in time AND space Finite Element methods

Motivation: Important BVPs and IBVPs in Engineering Science Elastostatics (example of an elliptic PDE) Elastodynamics (or the wave equation; example of a hyperbolic PDE) The heat equation (a parabolic PDE) Advective/diffusive behavior (example of a system which behaves like combinations of the above)

Elastostatics Partial differential equation corresponds physically to balance of linear momentum of a differential cube, with no inertial terms present Three dimensional statement of boundary value problem (BVP) requires two types (at least) of boundary conditions: force (Neumann) b.c.’s, and displacement (Dirichlet) b.c.’s One dimensional analogue is a simple rod or truss problem Classification of this system as an elliptic boundary value problem

Elastodynamics Obtained by restoring the inertial term to the elastostatic system Is an example of a (second order) Initial/Boundary Value Problem (IBVP), as both initial and boundary conditions are required to prescribe it One dimensional analogue is the wave equation Is the prototypical hyperbolic system (information propagates along characteristics)

Linear Heat Conduction (alternatively, mass diffusion) Governing equation represents energy (alternatively, mass) balance in a differential control volume Boundary conditions most often used specify flux (Neumann) or temperature (Dirichlet) Example of a parabolic differential equation (information travels infinitely fast)

Linear Advection/Diffusion An example of a system that displays characteristics of a hyperbolic system in one limit, and of a parabolic system in another Will be used in our discussion of numerics (homework) to demonstrate concepts of stability and accuracy (and their limits!)