1. Radiative Transfer with Predictor-Corrector Methods ABSTRACT TITLE : Radiative Transfer with Predictor-Corrector Methods OBJECTIVE: To increase efficiency, relative to traditional Monte Carlo methods, of Radiative Transfer computations METHODS: The “Gold Standard” Implicit Monte Carlo Method (IMC) is compared IMC with a predictor-corrector, and to the Carter-Forrest Method (CF) with and without predictor-corrector methods RESULTS: Both IMC and CF methods in 0-D showed error when applied traditionally and showed error with predictor-corrector methods. In the Monte Carlo results, the CF method maintained a error when reduced numbers of particles were used in the predictor step relative to the corrector step CONCLUSIONS: Although requiring more computational effort, the Carter-Forrest method appears equal to or superior to the IMC method in most numerical results at a larger computational cost. The IMC method has a bias in it when Alpha is not equal to 0.5 that will effect the accuracy of the numerical solution. Photon Teleportation does exist but Legendre-ME FET’s yield an improvement over traditional source tilting methods BACKGROUND OBJECTIVE The IMC and CF methods are studied to determine the accuracy of each method applied to the same problem. Next, both methods will be used in a predictor- corrector algorithm to determine the improvement of accuracy (decrease in error) of the methods when better approximations to the interaction cross sections are made during a time step. The goal is to improve the accuracy of the calculated solution or decrease the run time for the same accuracy. METHODS Solutions to the transport equation  Implicit Monte Carlo (IMC) and Carter-Forrest (CF) methods assume (Beta) and (sigma) are constants over a time step  IMC makes two more approximations that increase the error in the numerical solution yielding the differential form where -The main source of error for IMC comes from making constant over a time step and the two approximations stated previously. -Each approximation adds error into the approximation finally yielding the transport equation  CF makes only the constant and approximation in a time step and therefore has fewer sources of error but still retains the same size of error in the differential form -The main source of error for CF comes from making constant over a time step. The final transport form of the equation becomes where  Both IMC and CF are locally and globally with their approximations. Traditionally implementing these equations one would not expect any difference in order of decrease in error with time-step Predictor-Corrector Method  Our predictor-corrector method runs a Monte-Carlo calculation of a time step twice, once to predict the end of time step temperature (which yields cross section and solutions), then a second time to obtain results with that new information -Although doing twice the amount of work of a traditional method, predictor- corrector methods can significantly improve the accuracy of a calculation -In these transport problems, is updated with a predictor step to more accurately account for photon transport within a time step -Traditionally, the expansion of (sigma) is a basic Taylor Expansion -Using predictor-corrector methods, can now be expressed as a combination of the starting and predicted time steps yielding -Resulting in the situation that for twice the work, the order of error improves to second order -Variable Weight predictor correctors do not completely run the problem twice. By using fewer simulated particles with a higher weight, Variable Weight predictors attempt to get an approximate solution to use in the corrector step without doubling the overall computation requirements Wollaber-Larsen Cross Section Estimation  Create a time averaged cross section using the temperature estimation from the predictor step or  The Wollaber-Larsen Cross Section estimation reduces to the proposed predictor-corrector estimation for small time steps. 0-D RESULTS Problem Definition  Initial Temperature T = 0.1  = 1  Initial Photon Intensity (at time 0) = 1000*c* -Where c (speed of light) = 1 and is the starting Emissivity Relative Error vs. Time Step size for traditional IMC (lines) and CF (*)  The Radiation Transfer equations are shown below in their differential form The “Transport” Equation The “Material Energy” Balance Equations shown in two different forms The IMC method makes two fundamental approximations besides constant (Beta) and (sigma) over a time step 1) Emissions from the material occur instantaneously 2) The emissivity may be expressed in terms of the beginning and end of time step values The CF method only makes the constant and over a time step approximation. CF solves the transport equation exactly over a time step where IMC does not Temperature vs. Time for the Linear 0-D Problem Relative Error vs. Time Step size for Predictor-Corrector using Wollaber-Larsen Opacity correction with IMC (lines) and CF (*)  The temperature distribution of the Carter Forrest Method is exact  Implicit Monte Carlo will be warmer or colder depending on alpha during the transient  IMC and CF methods have “unphysical” solutions for large time steps ( time steps > )  When a sufficiently small time step is used, the error decreases as for both IMC and CF  IMC and CF methods no longer give “unphysical” solutions for large time steps  When a sufficiently small time step is used, the error decreases as for IMC and for CF For the following graphs, Blue line is at the same time, 10% into the problem Black line is at the same time, 50% into the problem Red line is at the same time, 100% into the problem 1-D RESULTS Problem Definition  Initial Temperature T = 0.1  = 1 /  = 0.56 *  1 KeV isotropic source at the left boundary with vacuum leakage Benchmark distribution Time Step (0.5) and Width (0.05)  Both the traditional approach and predictor-corrector (on the predictor step) violate the maximum principle  Even though the violations eventually dissipate, the waves that violated the maximum principle do not recreate the reference solution CONCLUSIONS  Predictor-Corrector methods give a better approximation to (sigma) and result in a more accurate solution for larger time step sizes  0-D Results show that the Carter-Forrest method may be turned into an method with predictor-corrector while Implicit Monte Carlo still turns out to be an method  The IMC Method has a bias in the transient based off of the value of Alpha that will occur in heating and cooling problems. This may help or hinder in the accuracy of the solution depending on other approximations  Photon teleportation is aggravated by the mean free paths of a cell and the time step size used to transport the radiation. Legendre-ME FET’s allow for a meaningful improvement over the current source tilting methods Jesse Cheatham, Dr. James Holloway, Dr. Bill Martin Problem Definition  Initial Temperature T = 400  = 1000 /  = 40*  Initial Photon Intensity (at time 0) = 1000*c* -Where c (speed of light) = 1 and is the starting Emissivity The IMC method has a bias in it that is related to the value of Alpha. By performing a residual error analysis of the analytic solutions for this linear problem, the bias can be seen as Alpha=1Alpha=0 Timing Test with Regard to Accuracy  The IMC predictor-corrector method is more accurate than the traditional approach at a large computational cost  A variable weight predictor corrector uses 1/100 th the number of particles in the predictor step than the corrector step. While gaining on over the traditional method, it is generally not as accurate as the full predictor corrector, but has great gains in speed. PHOTON TELEPORTATION  Photon Teleportation occurs due to the absorption-remission event and is dependent on the mean free path of a cell and the time step size used  Smaller time steps use to get to the same time increase the effect of photon teleportation for a cell greater than 1 mean free path  Using Legendre polynomials with Maximum Entropy correction, a much larger spatial width and refined time step can be used in this linear problem