Lacuna-based Artificial Boundary Condition And Uncertainty Quantification of the Two-Fluid Plasma Model Eder Sousa 1 , Uri Shumlak 1 and Guang Lin 2 1 Computational Plasma Dynamics Lab, University of Washington, Seattle, WA 2 Computational Sciences and Mathematics, Pacific Northwest National Laboratory, Richland, WA Abstract Modeling open boundaries is useful for truncating extended or infinite simulation domains to regions of greatest interest. However, artificial wave reflections at the boundaries can result for oblique wave intersections. The lacuna-based artificial boundary condition (ABC) method is applied to numerical simulations of the two-fluid plasma model on unbounded domains to avoid unphysical reflections. The method is temporally nonlocal and can handle arbitrary boundary shapes with no fitting needed nor accuracy loss. The algorithm is based on the presence of lacunae (aft fronts of the waves) in wave-type solutions in odd- dimensional space. The method is applied to Maxwell’s equations of the two-fluid model. Placing error bounds on numerical simulations results is important for accurate comparisons, therefore, the multi-level Monte Carlo method is used to quantify the uncertainty of the two-fluid plasma model as applied to the GEM magnetic reconnection problem to study the sensitivity of the problem to uncertainty on the mass ratio, speed of light to Alfven speed ratio and the magnitude of the magnetic field initial perturbation. Lacuna-based Artificial Boundary Conditions 3 I Numerical simulation of wave phenomena on unbounded domains often produce unphysical reflections from the boundaries I Consequently, The original infinite domain has to be truncated and special artificial boundary conditions (ABCs) have to be developed Lacunae are still regions present in wave-type solutions in odd-dimension spaces. Introduction I The key idea of using lacunae for computations is very simple: I If the sources of waves are compactly supported in space and time; I If the domain of interest has finite size; I Then it will completely fall inside the lacuna once a certain time interval has elapsed since the inception of the sources. I The lacunae-based ABC is nonlocal in space and time without loss of accuracy I The lacunae-based ABCs are not restricted geometrically to any particular shape of the external artificial boundary The computational domain and the auxiliary domain overlap by a couple grid cells where the transition multiplier, μ, varies from zero on the interior side to one on the exterior one. The three steps of ABC implementation: I Calculate the exterior source, Ω(v) , from the interior solution; I Reintegration of the exterior solution excluding earlier exterior sources I Communicate the exterior evolution with the interior problem ghost cells 3 S.V.Tsynkov, ”On the Application of Lacunae-based Methods to Maxwell’s Equations”, JCP 199 (2004) 126-149 Auxiliary source generation I The computational domain is advanced using: ∂ q ∂ t + ∇· F(q)= S (q) I Auxiliary problem: ∂ v ∂ t + ∇· F(v)= S (v) + Ω(q) I Ω(q) is the auxiliary source and v = μq I For non-moving boundaries there is no time dependent of μ(x), therefore: Ω(q)= μS (q) - S (μq)+ ∇· F(μq) - μ∇· F(q) The boundary formulation is applied to the Maxwell equations using the Washington Approximate Riemann Plasma Solver (WARPX). The field are modeled using the Perfectly Hyperbolic Maxwell 4 formulation to account for the divergence corrections, ∂ ~ B ∂ t + ∇× ~ E + γ ∇Ψ= 0 1 c 2 ∂ ~ E ∂ t -∇× ~ B + χ∇Φ= -μ o X s q s m s ρ s ~ u s 1 χ ∂ Φ ∂ t + ∇· ~ E = X s q s m s ρ s 1 γ c 2 ∂ Ψ ∂ t + ∇· ~ B = 0 Quantity being plotted is B z . – Interior domain boundary, – Auxiliary domain boundary Problem setup: a quarter of spherical pulse propagating outwards, where the left and the bottom boundary conditions are lacunae-based ABC’s. The wave front is communicated to the auxiliary problems by the auxiliary sources in the transition region (overlap region between the interior and the auxiliary regions). Exterior domain re-integration I The auxiliary sources drive the problem in the auxiliary domain and guarantees both solutions match in the exterior domain I The auxiliary problem is re-integrated every specified time interval and early sources are removed from computation The following plot are slices of the previous figures at x=1. Initially the interior problem pulse propagates through the interior domain. As the re-integration is preformed the earlier sources are removed from the auxiliary domain as they no longer affect the interior solution. As the interior pulse enters the transition region, the auxiliary source is applied to the auxiliary domain. The pulse is reintegrated out of the exterior domain an no reflection are present in the interior problem. Conclusion The lacuna-based ABC’s can correctly simulated unbounded domains accurately while removing artificial reflections that otherwise would be present. 4 Munz et. al., ”Divergence Correction Techniques for Maxwell Solvers Based on Hyperbolic Model”, JCP 161, 484-511 (2000) Uncertainty Quantification Motivation I Determining the region of acceptable results for experimental and computational is not only desired but required I Uncertainty quantification will allow for errorbars to be put into computational results I There are numerous sources of uncertainty in the Two-Fluid plasma model I Treating all the inputs as stochastic is computationally expensive Introduction I The mean square error (MSE) e ( ˆ P M ) 2 = V [ ˆ P MC m l ,N ]+(E [P m l - P ]) 2 I To achieve root MSE less than I the variance V [P MC m l ,N ]= N -1 V [P m l ] has to be less than 2 /2 meaning N ≥ -2 for the first term (large number of samples) I High discretization level: m l ≥ -1/α where α is the discretization convergence rate for the second term Multi-level Monte Carlo (MMC) 5,6 I The method is based on the multi-grid method as the solution is obtained from different solutions at different grid refinement levels I The estimator comes from the same random sample, N, but at different refinement levels, L ˆ P ML m l = L X l =0 1 N l N l X i =1 (P i m l - P i m l -1 ) I The multilevel variance V [Y l ]= V [P m l - P m l -1 ] → 0 as l →∞ which implies that N l → 1 as l →∞ I It is less costly to achieve an overall RMSE of for the multilevel than the standard Monte Carlo Probabilistic Collocation Method (PCM) I The PCM approach based on selecting the sampling points and corresponding weights. I Collocation points in probability space of random parameters as independent random inputs based on a quadrature formula I The solution statistics is estimated using the corresponding quadrature rule 5 M.B.Giles, ”Multilevel Monte Carlo Path Simulation”, Operations Research, 56, 981-986, 2008 6 K.A.Cliffe, et. al. ”Multilevel Monte Carlo Methods and Applications to Elliptic PDEs with Random Coefficients.” Submitted, to appear in Numerische Mathematik, 2011 Results The methods are applied to the two-fluid magnetic reconnection problem for the cases where the electron-to-ion mass ratio, the speed of light and the initial B-field perturbation are considered stochastic. Log M of the variance and mean for P l and P l - P l -1 (velocity at x=0) for the Euler equations with dispersive source term. The variance and the mean converge as the refinement level is increased. Mean (top) and variance (bottom) of the reconnected flux for varying speed of light to Alfv ´ en speed ratio ranging from 10 to 20. Reconnected flux for different plasma models used in the GEM challenge 7 compared to two-fluid model (red). The error bars are the standard deviation caused by varying the c /v A ratio, calculated using the MMC method. Mean and variance of the reconnected flux for varying electron-to-ion mass ratio ranging from 25 to 100. Mean and variance of the reconnected flux for varying amplitude of the B-field perturbation ranging from 8.5% to 11.5% of the background field. Computational Cost The following are actual computational cost for the case of the uncertainty in the initial B-field perturbation. CPU-hours MC 6496 PCM 4660 MMC 4075 Conclusion The MMC method produced the same accuracy as the standard MC and the PCM. A 37.3% cost saving was calculated for the MMC method over the MC and a 14.4% saving over the PCM. MMC allows for an easy and inexpensive way to determine the error of computational plasma codes. 7 ”Geospace Environmental Modeling (GEM) Magnetic Reconnection Challenge,” Journal of Geophysical Research, vol. 106, pp. 3715-3719, March 2001. Computational Plasma Dynamics Lab - Aeronautics and Astronautics Department - University of Washington - Seattle, WA [email protected] http://www.aa.washington.edu/research/cfdlab