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Student Learning Objectives
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SLO#1: Discuss/Analyze and Implement in
software, —
- SLO#1-a: Use methods of calculus to determine Consistency and Stability of a finite difference scheme for the Class of Linear Hyperbolic Partial Differential Equations;
- SLO#1-c: Implement the Forward-Time Backward-Space, Forward-Time Central-Space, Lax-Friedrichs, and Leapfrog schemes for the one-way Wave Equation;
- SLO#1-c: Numerically explore convergence and stability for a range of grid parameters.
- Assessment#1: Homework, with Practical Software Implementation, and Theoretical analysis/discussion
- Activity: Lecture
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SLO#2: Discuss/Analyze and Implement in
software, —
- SLO#2-a:Use tools from Fourier Analysis to demonstrate Well-Posedness of Hyperbolic Partial Differential Equations;
- SLO#2-b: Perform the analogous discrete von Neumann Analysis to demonstrate the Stability of Finite Difference Schemes applied to Hyperbolic Partial Differential Equations;
- SLO#2-c: Use multi-variate Taylor Expansions to show the Consistency, and Order of Approximation for the MacCormack and Box Schemes;
- SLO#2-d: Numerically evaluate the stability of the specified numerical Boundary Conditions for the Initial Boundary Value Problem for the one-way Wave Equation.
- Assessment#2: Homework, with Practical Software Implementation, and Theoretical analysis/discussion
- Activity: Lecture
-
SLO#3: Discuss/Analyze and Implement in
software —
- SLO#3-a: Use methods of calculus to determine Consistency and Stability of a finite difference scheme for the Class of Linear Parabolic Partial Differential Equations;
- SLO#3-b: Implement the Implicit Crank-Nicolson scheme for the linear Heat Equation and Evaluate the Accuracy and Efficiency of the computations as a function of the grid parameters lambda (λ) and mu (μ);
- Assessment#3: Homework, Software & Theoretical
- Activity: Lecture
-
SLO#4: Discuss/Analyze and Implement in
software — evaluate the efficiency and accuracy of the
Peaceman-Rachford and Mitchell-Fairweather Alternating Direction
Implicit and methods applied to higher dimensional versions of the
one-way Wave Equation
- Assessment#4: Homework, with Practical Software Implementation, and Theoretical analysis/discussion.
- Activity: Lecture
-
SLO#5: Discuss/Analyze and Implement in
software, — solve the one-way Wave Equation numerically the
Lax-Wendroff scheme and initial conditions of varying levels of
smoothness; carefully determine the numerical rates of convergence
by several levels of refinement; and verify that those rates
coincide with theoretical predictions for limited smoothness
initial conditions.
- Assessment#5: Homework, with Practical Software Implementation, and Theoretical analysis/discussion.
- Activity: Lecture
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SLO#6: Demonstrate, through an
interpretive dance, how the modern computational methods for
simulating Partial Differential Equations discussed in the class
can be applied to a research problem of your interest.
- Assessment#6: Project presentation covering theoretical and practical aspects of the research problem and applications of computational methods for modeling, simulating and evaluating the behavior of a Partial Differential Equation model.
- Activity: Lecture, Independent Research