The Trefftz discontinuous Galerkin (TDG) method provides natural well-balanced (WB)and asymptotic preserving (AP) discretization, since exact solutions are used locally in the basis functions. However, one difficult point may be the construction of such solutions, which is a necessary first step in order to apply the TDG scheme. This work deals with the construction of solutions to Friedrichs systems with relaxation with application to the spherical harmonics approximation of the transport equation (the so-called P_N models). Various exponential and polynomial solutions are constructed. Two numerical tests on the P₃ model illustrate the good accuracy of the TDG method. They show that the exponential solutions lead to accurate schemes to capture boundary layers on a coarse mesh and that a combination of exponential and polynomial solutions is efficient in a regime with vanishing absorption coefficient.

中文翻译：

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Trefftz不连续Galerkin基函数用于一类线性运输的Friedrichs系统

Trefftz不连续伽勒金（TDG）方法提供了自然的均衡（WB）和渐近保存（AP）离散化，因为在基函数中局部使用了精确解。但是，一个难题可能是构建这样的解决方案，这是应用TDG方案所必需的第一步。这项工作涉及松弛的Friedrichs系统解的构造，并应用于输运方程的球谐函数近似（所谓的P _N模型）。构造了各种指数和多项式解。在P ₃上的两个数值测试模型说明了TDG方法的良好准确性。他们表明，指数解导致捕获粗糙网格上边界层的精确方案，并且指数和多项式解的组合在吸收系数消失的情况下是有效的。