作为一类重要的偏微分方程组，双曲松弛系统描述了各种非平衡物理过程，具有广泛的应用背景。当实际的物理过程发生在带有边界的区域中时，给相应的双曲松弛系统提供合适的边界条件是必不可少的。然而一般来讲，很难直接从物理上获取正确的边界条件，原因是很多松弛系统的某些未知函数（比如高阶矩封闭系统中的高阶矩）本身就没有清楚的物理含义。我们试图从数学的角度为松弛系统构造合理的边界条件，并进行理论分析和数值验证。本文主要关注两个具体的双曲松弛模型，希望从中得到构造边界条件的方法，为进一步的研究工作提供思路。 本文首先研究一维线性化的Jin-Xin松弛模型，该模型是双曲守恒律方程的一种半线性近似。假设守恒律的边界条件已知，我们借助渐近分析为Jin-Xin模型构造了合理的边界条件。需要指出的是，这样构造的边界条件高度不唯一，含有一些自由参数。然后证明了部分边界条件满足广义Kreiss条件（保证双曲松弛系统初边值问题稳定的重要条件），即利用广义Kreiss条件作为判断准则来限制参数的范围。随后，通过分析Jin-Xin模型初始条件和边界条件的相容性，对参数做进一步的限制。紧接着，估计了Jin-Xin模型初边值问题精确解和近似解的误差。该误差蕴含着所构造初边值问题的精确解收敛到原守恒律方程初边值问题的精确解。最后，通过数值实验验证了理论分析的结果。 除了上述工作，本文还导出了Boltzmann方程的近似模型-BGK模型的一种积分型矩封闭系统的边界条件，分析了其数目的正确性，验证了所得到的边界条件满足Kreiss条件（保证双曲方程初边值问题适定的重要条件）。最后，通过数值实验证实了所得边界条件的有效性。

As an important class of partial differential equations, hyperbolic relaxation systems describe various non-equilibrium physical processes and have a wide range of applications. When actual physical processes occur in regions with boundaries, it is necessary to provide appropriate boundary conditions for the corresponding hyperbolic relaxation systems. However, it is generally difficult to obtain the correct boundary conditions from physical principles, because some unknown functions in many relaxation systems (such as the higher-order moments in higher order moment closure systems) do not have a clear physical meaning. We attempt to construct reasonable boundary conditions for relaxation systems from a mathematical perspective, supplemented with theoretical analysis and numerical verification. This thesis primarily focuses on two specific hyperbolic relaxation models, with the aim of obtaining methods of constructing boundary conditions and providing ideas for further work. To begin with, the thesis focuses on the one-dimensional linearized Jin-Xin relaxation model, which is a semi-linear approximation of the hyperbolic conservation laws. Assuming the boundary conditions of the hyperbolic conservation laws are given, we construct reasonable boundary conditions for the Jin-Xin model with the help of asymptotic analysis. It should be pointed out that the boundary conditions constructed in this way are highly non-unique and contain some free parameters. Then we partly show that they fulfill the generalized Kreiss condition (important conditions for ensuring the stability of initial-boundary value problem for hyperbolic relaxation systems). That is, we use the generalized Kreiss condition as a criterion to constrain the range of parameters. Furthermore, by analyzing the compatibility of the initial and boundary conditions of the Jin-Xin model, further restrictions are imposed on these parameters. Subsequently, the error between the exact solution and approximate solution to the initial-boundary value problem of the Jin-Xin relaxation model is estimated. This error implies that the exact solution to the constructed initial-boundary value problem converges to the exact solution to the initial-boundary value problem of the original conservation laws. Finally, we verify the results of theoretical analysis through numerical experiments. In addition to the aforementioned work, this thesis also derives the boundary conditions for a kind of quadrature-based moment closure systems of the BGK model, which is an approximate model of the Boltzmann equation. Then we analyze the correctness of their number and verify that the obtained boundary conditions satisfy the Kreiss condition (important conditions for ensuring the well-posedness of initial-boundary value problem for hyperbolic equations). Lastly, the effectiveness of the obtained boundary conditions is verified through numerical experiments.