1、Introduction to Numerical Methods for ODEs and PDEs,Methods of Approximation Lecture 3: finite differences Lecture 4: finite elements,Prevalent numerical methods in engineering and the sciences,We will introduce in some detail the basic ideas associated with two classes of numerical methods Finite D
2、ifference Methods (in which the strong form of the boundary value problem, introduced in the model problems, is directly approximated using difference operators) Finite Element Methods (in which the weak form of the boundary value problem, derived through integral weighting of the BVP, is approximat
3、ed instead) .while skipping a third class of methods which are quite prevalent Boundary Element Methods (BEM) Predominantly for linear problems; based on reciprocity theorems and Greens function solutions,Finite Difference Methods,Rely on direct approximation of governing differential equations, usi
4、ng numerical differentiation formulas Ordinary derivative approximations Forward difference approximations Backward difference approximations Central difference operators Partial derivative approximations,Applications of finite differencing strategies,Time integration of canonical initial value prob
5、lems (ODEs) Stability and accuracy; unconditional versus conditional stability Implicit vs. explicit schemes Finite difference treatment of boundary value problems (steady state) Case study: 1D steady state advection-diffusion Stabilization through upwinding,Applications of finite differencing strat
6、egies (cont.),Finite difference treatment of initial/boundary value problems (time and space dependent) Semi-discrete approaches (method of lines),Finite Element Methods,Using the 1D rod problem (elliptic) as a template: Development of weak form (variational principle) Galerkin approximation versus other weighting approaches Development of discrete equations for linear shape function case,
