Experimental Mathematics and Benchmark Solutions in Transport and Diffusion Theory

Finding benchmark solutions to the equations of neutron transport or diffusion theory is an experimental science. After all, we choose a numerical method, test the method, usually through sensitivity studies, and, based on our best information concerning numerical error, fix the numerical parameters and/or develop adaptive strategies. Regarded as a sensitivity study, convergence acceleration is an experimental approach that has the potential to greatly enhance the precision of transport solutions. It represents a transformative way to redefine the essence of a numerical solution. The fundamental idea is to consider the true solution as the limit of numerical approximations, such as found by discretization or iteration, and accelerate the sequence of approximations to its limit. Using the determined elements of a sequence, convergence acceleration converges the sequence by detecting its asymptotic behavior. Since there is no overall theory of convergence acceleration, it is an experimentally based numerical method. In my presentation, we will “discover” convergence acceleration through simple, and not so simple, examples covering a variety of solutions to the transport and diffusion equations. Time permitting, I will show how convergence acceleration provides highly precise solutions to the diffusion equation for criticality in 1D geometry as well as for discrete ordinates solutions for fixed sources in 1D slabs. My objective is to present most likely more than you wanted to know about convergence acceleration; but at least, you will be made aware of its potential to generate benchmarks and in the classroom.