INSTRUCTOR: J. K. Shultis, Office WD-125, (913) 532-5626; email jks@ksu.edu

PREREQUISITES: Knowledge of (1) basic heat transfer and fluid flow, (2) calculus through partial differential equations, (3) neutron diffusion theory, (4) basic numerical methods, (5) a programming language.

COURSE GOALS: (1) Understand many different numerical approximations used to solve the equations of heat and fluid flow. (2) Understand the mathematical theory behind standard iterative methods for linear algebraic equations, (3) appreciate the use of adjoint techniques, and (4) understand modern algorithms for solution of the hydrodynamic equations.

REFERENCE TEXTS:

• S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980.
• S. Nakamura, Computational Methods in Engineering and Science, Wiley, New York, 1977.
• J. Crank, The Mathematics of Diffusion, Clarendon Press, Oxford, 1975.
• P.E. Allaire, Basics of the Finite Element Method, Wm Brown Publ.
  

TOPICS:

1. Classification of Second-Order Partial Differential Equations:

• Properties and solution methods for elliptic, parabolic, and hyperbolic pde's.
• Types of boundary conditions: over, under and uniquely specified problems.
• Examples of analytical solutions.

2. Theory of Heat Conduction:

• Fourier's law of heat conduction; thermal conductivity.
• Derivation of the transient heat conduction equation.
• Need for a hyperbolic form of heat conduction equation; analogy to generalized Fick's law in neutron diffusion theory.

3. Finite-Difference Approximations:

• Derivation of FD equations for steady-state heat conduction (elliptic pde) in 1, 2 and 3 dimensions.
• General boundary condition approximations.
• Use of tridiagonal matrix algorithm for 1-D problems.
• Iteration algorithms (Jacobic, Gauss-Seidel, successive-over-relaxation, ADI, two-cycle Jacobi, strongly implicit methods).

4. Theoretical Aspects of Iterative Methods:

• Review of matrix eigenvalue properties (degenerate and non-degenerate cases)

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• Convergence and convergence rate of iterative algorithms.
• Gerschgoerin's theorem and its consequences.
• Properties of Jacobi iteration matrix.
• Properties of relaxation iteration matrices; optimal relaxation parameters; methods for estimating optimum relaxation parameter.

5. Numerical Solution of Hydrodynamic Equations:

• Finite difference approximation and boundary equations.
• Central differences, upwind, linear upwind, and quadratic upwind approximations.
• The need for staggered grids in momentum equations.
• Methods for estimating the pressure field: SIMPLE and SIMPLER algorithms.

6. Time-dependent Flow Fields:

• Implicit, explicit, and Crank-Nicholson time discretization methods.
• Truncation error, accuracy, and iterative stability.
• Von Neumann stability analysis.

7. Finite-Element Method:

• Weighted residual approach for discretization of flow equations.
• Collocation, region balancing, and Galerkin approximations.
• Application of 2-D triangular elements to steady-state heat conduction.
• Grid generation and numbering.

8. Iterative Acceleration Techniques:

• Need for acceleration methods.
• Chebychev one- and two-parameter methods.
• Coarse mesh rebalancing methods; additive and multiplicative rebalancing, decoupling coefficients.

9. Monte Carlo Methods:

• Multi-dimensional integration with Monte Carlo.
• Solving linear algebraic equations by a random walk problem.
• Application to steady-state heat conduction.
• Advantage of adjoint formulations.
• Discrete versus continuous random walk approaches.
• Accelerating the random walk simulation.