An Investigation of the Numerical Methods of Finite Differences and Finite Elements for Digital Computer Solution of the Transient Heat Conduction (Diffusion) Equation Using Optimum Implicit Formulations.
AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OHIO SCHOOL OF ENGINEERING
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The transient heat conduction equation, with Dirichlet and Neumann boundary conditions, is solved by the methods of finite-differences and finite-elements, and the numerical solutions are investigated with respect to accuracy and stability. A general six point finite-difference expression is used for which there exists a high order accurate modification. The finite-element method used is based on a stationary variational principle. Several methods for treating accuracy and convergence problems which result from a discontinuity in the initial condition are investigated. The Crank-Nicolson method is a special case of both the finite-difference and finite-element methods. The finite-difference version of the Crank-Nicolson method is shown to be more accurate than the finite-element version, especially when a discontinuity exists between the initial condition and the boundary conditions. The high order accurate schemes for both finite-differences and finite-elements are shown to be equivalent for the case of linear elements. Some of the results suggest the possibility of finding a finite-element scheme which is highly accurate in a mean square sensr over the entire solution domain. Author