Discontinuous Spectral Element Method For Solving Radiative Transfer Equation In Cylindrical Coordinate System

The discontinuous spectral element method(DSEM)is extended to solve the discrete ordinates form of radiative transfer equation(RTE)in axisymmetric cylindrical enclosures.The DSEM combines the traits of the discontinuous finite element method(DFEM)and the spectral methods.It permits the discontinuity of variables across the element boundaries and can offer h and p convergence,i.e.,the accuracy of the results can be improved by increasing the element number and the polynomial order respectively.The orthogonal polynomials are employed to construct the nodal basis functions,and the piecewise constant angular(PCA)quadrature is used for the angular discretization.The performance of the DSEM is investigated by comparing convergence properties and CPU time.Results show that for 1-D problems,its p-convergence rate is very fast,following the exponential law,but its hconvergence rate is only second-order.For 2-D problems,the p and h convergence rates drop to second and first order respectively.Considering the CPU time,the DSEM shows advantage over the discrete ordinates method(DOM)only for 1-D problems,while,for 2-D problems,its performance is something worse than that of DOM.For the problem with discontinuous boundary conditions,the results of DSEM suffer from severe ray effect when the number of discrete directions is inadequate,similar to DOM.By applying angular refinement,the ray effect can be mitigated obviously.For such problems,increasing the accuracy of results needs refining the spatial and angular grids simultaneously.For the problem with moderate complex geometry,the DSEM can obtain accurate solutions and is quite suitable for such problems.Finally,the spectral method is also investigated for supplement.The results show that the convergence properties of the two methods are basically the same: their convergence rates both follow the exponential law for 1-D problems,but drop to second-order for 2-D problems.