
Students will see some practical applications of numerical methods.
Compare explicit and implicit Euler methods, derived from a Taylor series perspective, to approximate y at t+Δt using the slope at t and the challenge of next-step values.
Explore general Runge-Kutta formulations for second-order methods, using k1 and k2 slope estimates and weighted averages to advance y, with examples like modified Euler, midpoint rule, and Ralston's method.
Introduces boundary value problems by deriving the steady-state temperature distribution along a one-dimensional cooling fan, leading to a second-order elliptic equation with end boundary conditions solved by the shooting method.
This lecture replaces the right boundary with a derivative condition at x=1, using a central difference and a ghost point to form a tridiagonal system solved by the tdma algorithm.
This lecture shows a student how to set up a ParaView case for the flow about a cylinder.
Learn to non-dimensionalize differential equations by using scale constants for temperature, time, and space in a two-dimensional diffusion problem, then back-substitute t_bar, x_bar, y_bar for real time and length.
The course provides an introduction to the numerical solution of ordinary and partial differential equations and is at a level appropriate for undergraduate-level STEM students. Prior knowledge of numerical methods is helpful but not necessary as (most) prerequisite material is introduced on an as-needed basis. Knowledge of a scientific programming language is necessary for those wishing to write their own codes. All codes used to demonstrate methods and solve example problems (primarily in both Fortran and Python) are available for downloading, as are the class notes. For the ordinary differential equations, we will study numerical techniques to solve:
1) Initial value (or propagation) problems
2) Boundary value (or equilibrium) problems
3) Eigenvalue (or characteristic value) problems
In terms of partial differential equations, we will concentrate on finite-difference approaches to solve second-order partial differential equations.
These equations may be classified as elliptic, parabolic, or hyperbolic. The classification helps determine the best approach to obtain a numerical solution. We will focus on elliptic and parabolic partial differential equations.
The primary course sections are:
SECTION 2: ODE’s: INITIAL VALUE PROBLEMS
SECTION 3: ODE’s: BOUNDARY VALUE PROBLEMS
SECTION 4: ODE’s: EIGENVALUE PROBLEMS
SECTION 5: ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
SECTION 6: PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS