Spatial convergence study of Lax-Friedrichs WENO fast sweeping method on the SN transport equation with nonsmoothness

Abstract Spatial convergence study of the Lax-Friedrichs fast sweeping method based on the classical third order WENO scheme (WENO3) without a priori smoothness indicators is performed on the variants of Larsen’s benchmark problem with nonsmooth solutions. For comparison, third order upwind (UPWD3), weighted diamond difference, step method, bilinear discontinuous finite element method are also included. In optically thick regimes of the cases with continuous angular fluxes, WENO3 detects steep gradients of angular fluxes and handles them as discontinuities to suppress the unphysical oscillation, which improves the accuracy significantly compared with UPWD3. In the cases with discontinuous angular fluxes, WENO3 achieves non-oscillatory solutions with limited numerical diffusion. Nonlinear Gauss-Seidel iteration is necessary to obtain convergent solution for the nonlinearity of WENO3, which leads to low computational efficiency. The convergence behavior of nonlinear Gauss-Seidel iteration is affected by nonsmoothness of the exact solution, mesh number and the relaxation parameter.

[1]  J. M. Powers,et al.  Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points , 2005 .

[2]  Dean Wang,et al.  High-Order Lax-Friedrichs WENO Fast Sweeping Methods for the SN Neutron Transport Equation , 2019, Nuclear Science and Engineering.

[3]  Marvin L. Adams,et al.  Discontinuous Finite Element Transport Solutions in Thick Diffusive Problems , 2001 .

[4]  Analysis of inherent oscillations in multidimensional SN solutions of the neutron transport equation , 1996 .

[5]  W. W. Engle,et al.  New weighted-difference formulation for discrete-ordinates calculations , 1977 .

[6]  Jean C. Ragusa,et al.  On the Convergence of DGFEM Applied to the Discrete Ordinates Transport Equation for Structured and Unstructured Triangular Meshes , 2009 .

[7]  Yousry Y. Azmy,et al.  Spatial Convergence Study of Discrete Ordinates Methods Via the Singular Characteristic Tracking Algorithm , 2009 .

[8]  Edward W. Larsen,et al.  Advances in Discrete-Ordinates Methodology , 2010 .

[9]  E. Lewis,et al.  Computational Methods of Neutron Transport , 1993 .

[10]  H. Huynh,et al.  Accurate Monotonicity-Preserving Schemes with Runge-Kutta Time Stepping , 1997 .

[11]  Marvin L. Adams,et al.  Characteristic methods in thick diffusive problems , 1998 .

[12]  Weitao Chen,et al.  Lax-Friedrichs fast sweeping methods for steady state problems for hyperbolic conservation laws , 2013, J. Comput. Phys..

[13]  Chao Yang,et al.  JSNT-S: A Parallel 3D Discrete Ordinates Radiation Transport Code on Structured Mesh , 2018, Volume 4: Nuclear Safety, Security, and Cyber Security; Computer Code Verification and Validation.

[14]  Chi-Wang Shu,et al.  High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments , 2016, J. Comput. Phys..

[15]  Wai-Sun Don,et al.  An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws , 2008, J. Comput. Phys..

[16]  Yousry Y. Azmy,et al.  Error Comparison of Diamond Difference, Nodal, and Characteristic Methods for Solving Multidimensional Transport Problems with the Discrete Ordinates Approximation , 2007 .

[17]  Y. Y. Azmy.  Arbitrarily high order characteristic methods for solving the neutron transport equation , 1992 .

[18]  Edward W. Larsen.  Spatial Convergence Properties of the Diamond Difference Method in x,y Geometry , 1982 .

[19]  Y. Y. Azmy,et al.  The Weighted Diamond-Difference Form of Nodal Transport Methods , 1988 .

[20]  Yousry Y. Azmy,et al.  Comparison of Spatial Discretization Methods for Solving the SN Equations Using a Three-Dimensional Method of Manufactured Solutions Benchmark Suite with Escalating Order of Nonsmoothness , 2015 .

[21]  Tangpei Cheng,et al.  Development of 3-D parallel first-collision source method for discrete ordinate code JSNT-S , 2020 .

[22]  Kirk A. Mathews,et al.  On the propagation of rays in discrete ordinates , 1999 .

[23]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[24]  R. D. Lawrence.  Three-dimensional nodal diffusion and transport methods for the analysis of fast-reactor critical experiments☆ , 1986 .

[25]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes , 2020, Acta Numerica.

[26]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[27]  Hongchun Wu,et al.  A nodal SN transport method for three-dimensional triangular-z geometry , 2007 .