Abstract

The widely used AZURV1 transport benchmarks package provides a suite of solutions to isotropic scattering transport problems with a variety of initial conditions. Most of these solutions have an initial condition that is a Dirac delta function in space; as a result these benchmarks are challenging problems to use for verification tests in computer codes. Nevertheless, approximating a delta function in simulation often leads to low orders of convergence and the inability to test the convergence of high-order numerical methods. While there are examples in the literature of integration of these solutions as Green’s functions for the transport operator to produce results for more easily simulated sources, they are limited in scope and briefly explained. For a sampling of initial conditions and sources, we present solutions for the uncollided and collided scalar flux to facilitate accurate testing of source treatment in numerical solvers. The solution for the uncollided scalar flux is found in analytic form for some sources. Since integrating the Green’s functions is often nontrivial, discussion of integration difficulty and workarounds to find convergent integrals is included. Additionally, our uncollided solutions can be used as source terms in verification studies, in a similar way to the method of manufactured solutions.

摘要

广泛使用的 AZURV1 传输基准程序包为具有各种初始条件的各向同性散射传输问题提供了一套解决方案。这些解决方案中的大多数都有一个初始条件，即空间中的狄拉克函数。因此，这些基准是用于计算机代码验证测试的具有挑战性的问题。然而，在模拟中逼近 delta 函数通常会导致低阶收敛，并且无法测试高阶数值方法的收敛性。虽然文献中有将这些解决方案集成为运输运营商的格林函数的示例，以便为更容易模拟的源生成结果，但它们的范围有限并简要说明。对于初始条件和来源的抽样，我们提出了未碰撞和碰撞标量通量的解决方案，以促进在数值求解器中准确测试源处理。对于某些来源，以解析形式找到了未碰撞标量通量的解决方案。由于积分格林函数通常很重要，因此讨论了积分难度和寻找收敛积分的解决方法。此外，我们的未冲突解决方案可以用作验证研究中的源术语，其方式类似于制造解决方案的方法。包括关于积分难度和寻找收敛积分的变通方法的讨论。此外，我们的未冲突解决方案可以用作验证研究中的源术语，其方式类似于制造解决方案的方法。包括关于积分难度和寻找收敛积分的变通方法的讨论。此外，我们的未冲突解决方案可以用作验证研究中的源术语，其方式类似于制造解决方案的方法。