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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 Difference 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
approximated 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 Green’s function solutions
Finite Difference Methods
Rely on direct approximation of governing
differential equations, using numerical
differentiation formulas
• Ordinary derivative approximations
– Forward difference approximations
– Backward difference approximations
– Central difference operators
• Partial derivative approximations
Applications of finite differencing
strategies
1. Time integration of canonical initial value
problems (ODEs)
•
•
Stability and accuracy; unconditional versus
conditional stability
Implicit vs. explicit schemes
2. Finite difference treatment of boundary value
problems (steady state)
•
•
Case study: 1D steady state advection-diffusion
Stabilization through upwinding
Applications of finite differencing
strategies (cont.)
3. 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