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Parallel Jacobian-free Newton Krylov solution of the discrete ordinates method with flux limiters for 3D radiative transfer. (English) Zbl 1259.78048
A radiative transfer problem is considered in the form of an integro-differential equation with boundary conditions. In [W. F. Godoy and P. E. DesJardin, J. Comput. Phys. 229, No. 9, 3189–3213 (2010; Zbl 1187.65143)], the discrete ordinates method (DOM) and finite volume methods (FVMs) with flux limiters were applied to discretise the angular domain and the spatial domain, respectively. A Newton iteration yields the numerical solution of the resulting nonlinear systems of algebraic equations, where the linear systems are solved iteratively by the generalised minimal residual (GMRES) method, i.e., a Newton-Krylov technique is investigated. In the subsequent paper, W. F. Godoy and X. Liu consider Jacobian-free methods within the GMRES iteration to save memory and to reduce the computational effort.
In the GMRES method, the Jacobian matrix is avoided by formulating the required matrix-vector product as a derivative. The derivatives are either calculated analytically (semi-exact approach) or computed by numerical differentiation. The latter technique requires the application of smooth flux limiters. The authors investigate both the Gram-Schmidt algorithm and the Householder transformation for the required orthogonalisation in the Newton-Krylov method. The construction of appropriate preconditioners is not within the scope of this paper.
About half of the paper consists in the presentation of two test examples, in which the efficiency of the Jacobian-free approach is investigated in detail. In each example, the authors compare results for the step limiter and the van Leer limiter. The first example is a three-dimensional homogeneous isotropic scattering medium. The optimal choice of the increment within the numerical differentiation, which is used to avoid a computation of the Jacobian matrix, is examined. The second example is a three-dimensional non-homogeneous pure scattering medium modelling a stratocumulus cloud. Now the authors investigate a combined memory-shared and memory-distributed parallelisation using the software library MPI on a supercomputer. Spatial domain decompositions are employed for this purpose. The tests involve up to 2048 CPU cores and also different numbers of threads. The results demonstrate impressive speed-ups, and thus parallel efficiency is achieved.

78M25 Numerical methods in optics (MSC2010)
65F10 Iterative numerical methods for linear systems
65H10 Numerical computation of solutions to systems of equations
65R20 Numerical methods for integral equations
65Y05 Parallel numerical computation
78A40 Waves and radiation in optics and electromagnetic theory
78A45 Diffraction, scattering
Full Text: DOI
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