The flux vector splitting (FVS) method has firstly been incorporated into the discontinuous Galerkin (DG) framework for reconstructing the numerical fluxes required for the spatial semi-discrete formulation, setting it apart from the conventional DG approaches that typically utilize the Lax-Friedrichs flux scheme or classical Riemann solvers. The control equations of hyperbolic conservation systems are initially reformulated into a flux-split form. Subsequently, a variational approach is applied to this flux-split form, from which a DG spatial semi-discrete scheme based on FVS is derived. In order to suppress numerical pseudo-oscillations, the smoothness measurement function IS from the WENO limiter is integrated into the TVB(D)-minmod limiter, constructing an optimization problem based on the smoothness factor constraint, thereby realizing a TVB(D)-minmod limiter applicable to arbitrary high-order polynomial approximation. Subsequently, drawing on the ``reconstructed polynomial and the original high-order scheme's L2 -error constraint'' from the literature [1] , combined with our smoothness factor constraint, a bi-objective optimization problem is formulated to enable the TVB(D)-minmod limiter to balance oscillation suppression and high precision. As for hyperbolic conservation systems, limiters are typically required to be used in conjunction with local characteristic decomposition. To transform polynomials from the physical space to the characteristic space, an interpolation-based characteristic transformation scheme has been proposed, and its equivalence with the original moment characteristic transformation has been demonstrated in one-dimensional scenarios. Finally, the concept of ``flux vector splitting based on Jacobian eigenvalue decomposition'' has been applied to the conservative linear scalar transport equations and the nonlinear Burgers' equation.

翻译：本文首次将通量矢量分裂（FVS）方法引入间断Galerkin（DG）框架，用于重构空间半离散格式所需的数值通量，这区别于通常采用Lax-Friedrichs通量格式或经典Riemann求解器的传统DG方法。首先将双曲守恒系统的控制方程改写为通量分裂形式，随后对该形式应用变分方法，推导出基于FVS的DG空间半离散格式。为抑制数值伪振荡，将WENO限制器中的光滑度量函数IS融入TVB(D)-minmod限制器，构建基于光滑因子约束的优化问题，从而实现了适用于任意高阶多项式逼近的TVB(D)-minmod限制器。进一步，借鉴文献[1]中“重构多项式与原高阶格式L2误差约束”的思想，结合本文的光滑因子约束，构建了双目标优化问题，使TVB(D)-minmod限制器能在振荡抑制与高精度之间取得平衡。对于双曲守恒系统，限制器通常需结合局部特征分解使用。为将多项式从物理空间转换到特征空间，提出了一种基于插值的特征变换方案，并在一维情形下证明了其与原矩量特征变换的等价性。最后，将“基于Jacobian特征值分解的通量矢量分裂”概念应用于守恒型线性标量输运方程和非线性Burgers方程。