带源项双曲型守恒律中心格式的单调性、稳定性和构造（修订版）
On monotonicity, stability, and construction of central schemes for hyperbolic conservation laws with source terms (Revised Version)
摘要：
研究了一般双曲型守恒律方程组差分格式的单调性和稳定性。其基本方法是研究一个非线性格式相对于其相应的变分格式的稳定性和单调性。这种方法导致应用线性方程系统的稳定性理论来建立相应的非线性方案的稳定性。主要的方法创新是定理建立的概念，一个非线性计划是稳定的（和单调），如果相应的计划在变化是稳定的（和，分别，单调）。给出了偏微分方程数值分析中差分格式的单调性和稳定性的判别准则。推广了Friedrichs（1954）的定理，使之适用于非对称矩阵的变分格式。对中心Lax-Friedrichs（LxF）格式进行了二阶精度的改进。在中心格式中使用单调分段三次插值来精确逼近模型。研究了改进格式的稳定性和单调性。一些版本的修改后的计划进行了测试的几个守恒律，该计划被发现是准确和强大的。作为应用到双曲型守恒律，在一般情况下，刚性源项，它是基于算子分裂技术构造的二阶格式。
英文摘要：
The monotonicity and stability of difference schemes for, in general, hyperbolic systems of conservation laws with source terms are studied. The basic approach is to investigate the stability and monotonicity of a non-linear scheme in terms of its corresponding scheme in variations. Such an approach leads to application of the stability theory for linear equation systems to establish stability of the corresponding non-linear scheme. The main methodological innovation is the theorems establishing the notion that a non-linear scheme is stable (and monotone) if the corresponding scheme in variations is stable (and, respectively, monotone). Criteria are developed for monotonicity and stability of difference schemes associated with the numerical analysis of systems of partial differential equations. The theorem of Friedrichs (1954) is generalized to be applicable to variational schemes with non-symmetric matrices. A new modification of the central Lax-Friedrichs (LxF) scheme is developed to be of the second order accuracy. A monotone piecewise cubic interpolation is used in the central schemes to give an accurate approximation for the model in question. The stability and monotonicity of the modified scheme are investigated. Some versions of the modified scheme are tested on several conservation laws, and the scheme is found to be accurate and robust. As applied to hyperbolic conservation laws with, in general, stiff source terms, it is constructed a second order scheme based on operator-splitting techniques.
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