复杂流场中存在的各种间断结构, 如激波、接触间断和界面, 为高精度数值求 解方法的发展带来了各种不同的挑战性问题。加权基本无振荡 (Weighted Essentially Non-Oscillatory, WENO) 格式是近三十年来高精度激波捕捉格式的突出代 表, 但在光滑区域其内在的数值耗散常会掩盖真实的物理耗散, 对接触间断和界 面的分辨仍有待提高。近年来, 基于边界变差最小化 (Boundary Variation Diminishing, BVD) 及双曲正切函数的界面捕捉方法 (Tangent of Hyperbola for INterface Capturing, THINC) 由于具有较高的界面分辨率而在相关领域得到较大的关注。 本文通过发展相关的数学物理分析方法, 研究综合性能更好的高效高分辨率间断 捕捉算法。主要研究工作如下： (1) 推广发展了用于分析非线性格式频谱性质 (色散和耗散) 的近似色散关系 (Approximate Dispersion Relation, ADR) 方法, 该方法能有效抑制原方法分析非线 性格式产生的混淆误差。利用新方法研究了现有 THINC 格式的频谱特性, 除了 具有 TVD 格式的一些特性外, 如限制器形式和在极值点降阶等, 本文研究进一步 表明, 在双曲函数中引入的陡度参数, 其变化主要影响波数区域的中间部分。当 陡度参数大于临界值时, 保对称 THINC (SP-THINC) 格式将产生反耗散和反色散 性质; 而对保单调 THINC (MP-THINC) 格式, 陡度参数大到一定值后将从反耗散 转为正耗散。 (2) 提出和发展了基于一般 S 型函数的界面捕捉算法 (Sigmoid Functions for INterface Capturing, SFINC)。通过比较研究, 发现连续性的 S 型函数均可用于构 造界面值的重构方法, 其中双曲幅角函数具有更良好的综合性质 (无振荡及锐利 的间断解)。结合单级 BVD 原则, 发展了 SFINC-BVD 算法。与 WENO 和 THINCBVD 格式相比, 新方法提高了激波、膨胀波和接触间断的分辨率, 有效避免了间 断附近产生非物理解。 (3) 研究了黎曼求解器对多级 BVD 算法的影响。使用 HLL (Harten-Lax-van Leer) 和 AUSM (Advection Upstream Splitting Method) 黎曼求解器, 对多级 BVD 算 法进行比较研究。将 HLL 格式的对流项和压力项分开处理, 发现 HLL 格式的压 力项在亚声速区域引入了额外的数值耗散。将 AUSM 类格式作为近似黎曼求解 器能够更好地模拟极值点和接触间断, 展现更丰富的小尺度结构。 (4) 结合边界变差分析了几种典型黎曼求解器的耗散特征, 提出了修正的 BVD 算法: 即在原边界变差的计算中, 考虑马赫数效应及黎曼求解器中引入的 特征速度效应, 利用此变差最小化实现了新的算法。数值算例表明新算法能有效 抑制原 BVD 算法在实际问题中可能出现的非物理过冲现象。 (5) 提出和发展了可压缩欧拉方程局部特征场分解的混合重构方法。可压缩 欧拉方程的特征值系统, 其局部特征场可以分解为两个本质非线性场和其余的线 性退化场。根据两类特征场的数学物理性质, 本文提出对非线性场采用鲁棒性较好的 WENO 重构, 而对于线性退化场采用分辨率较高的 BVD 格式或其它低耗散 无振荡格式 (如 TENO 格式)。新方法既提高了接触间断和极值点的分辨率, 又能 有效抑制激波附近出现的振荡和过冲。 (6) 提出了一致三阶和一致四阶精度的两级 BVD 算法。将 PmTn-BVD (m 阶 多项式 n 级 BVD) 算法推广发展了三阶 P2T2-BVD, 与高阶 BVD 格式相比, 低阶 格式具有较好的稳定性和较强的鲁棒性, 但与三阶 WENO 格式类似, P2T2-BVD 格式在二阶临界点只有二阶精度。因此本文在 P4T2-BVD 格式的框架下, 提出 利用三点 2 阶多项式 (P2) 及四点 3 阶多项式 (P3) 重构取代第一级 BVD 选择中 的 P4 多项式重构, 获得了在前二阶临界点具有一致三阶精度的 (U3-BVD) 及前 三阶临界点具有一致四阶精度的 (U3-BVD) 二级 BVD 格式。与 P4T2-BVD 相比, U3-BVD 具有三阶迎风的性质, 具有较好的稳定性, 而 U4-BVD 则具有中心格式 的基本无耗散性质, 对高频波具有更好的分辨能力。 (7) 采用双曲正割函数代替 TENO 格式中的截断函数, 发展了耗散更低的三 阶 TENO 格式以及基于该格式的 TENO-T2-BVD 格式。不同于截断函数中完全 不考虑间断模板的贡献, 正割函数是一光滑函数, 通过适当的参数控制, 可以有效 增加次光滑模板的贡献, 从而能有效提高三阶 TENO 的分辨率。由于三阶 TENO 格式与 THINC 重构所使用的模板节点相同, 因此 TENO-T2-BVD 格式具有更好 的鲁棒性。通过数值算例验证了所发展算法的高精度高分辨率特性。 (8) 提出了一种三级四阶对角隐式龙格-库塔方法 (Diagonal Implicit RungeKutta, 简称 DIRK)。不但比传统的三级三阶显式龙格-库塔方法具有更大的稳定 区域, 可以求解带一定刚性的问题, 而且比已有的三级四阶对角隐式龙格-库塔方 法具有更高的精度。将该隐式时间积分方法与边界变差最小化算法结合, 线性对 流方程、非线性伯格斯方程和欧拉方程的数值结果都表明, 无论是在光滑区域, 还是在间断附近, 即便采用比较大的 CFL (Courant-Friedrichs-Lewy) 数仍然能够 获得稳定和精确的解。

Various discontinuity structures in complex flow fields, such as shock waves, contact discontinuities, and interfaces, pose various challenging problems for the development of high-accuracy numerical methods. The weighted essentially non-oscillatory (WENO) scheme is a prominent one of high-accuracy shock-capturing schemes in the past three decades. However, in smooth regions, its inherent numerical dissipation often overwhelms the pratical physical dissipation, and the resolution of contact discontinuities and interfaces still needs improving. In recent years, the interface-capturing schemes based on boundary variation diminishing (BVD) and tangent of hyperbola for interface capturing (THINC) algorithms have received significant attention in related fields due to their high resolution. This research aims to construct high-efficiency and high-resolution discontinuity-capturing algorithms with better comprehensive performance by developing relevant mathematical and physical analysis. The main work includes: (1) The approximate dispersion relation (ADR) method for analyzing the spectral properties (dispersion and dissipation) of nonlinear schemes is developed. The new method can effectively suppress the aliasing errors generated by nonlinear schemes. Then, it is used to study the spectral properties of existing THINC schemes, which have some features of TVD schemes, such as limiter form and order reduction at local extrema. This study shows that the steepness parameter introduced in the hyperbolic function mainly affects the middle part of the wavenumber region. When the steepness parameter is greater than the threshold, the symmetry-preserving THINC (SPTHINC) scheme will exhibit anti-dissipation and anti-dispersion properties; For the monotonicity-preserving THINC (MP-THINC) scheme when the steepness parameter is greater than a certain value, it loses the anti-dissipation property. (2) The SFINC (sigmoid function for interface capturing) scheme is proposed and developed based on general sigmoid functions. Through comparative research, it is found that continuous sigmoid functions can be used for interface reconstruction, among which the hyperbolic amplitude function has better comprehensive properties (essentially non-oscillation and sharp discontinuity). The SFINC-BVD scheme is developed based on the single-stage BVD algorithm. Compared with WENO and THINC-BVD schemes, the novel method not only significantly improves the resolution of shocks, rarefaction waves, and contact discontinuities, but also effectively avoids the non-physical solutions near discontinuities. (3) The influences of different Riemann solvers on the BVD algorithm are investigated. The Harten-Lax-van Leer (HLL) and advection upstream splitting method (AUSM) schemes are two kind of classical approximate Riemann solvers for solving the Euler eqations in CFD. In this work, a systematic comparison study of the PnTm-BVD (PnTm: Polynomial of n-degree and THINC function of m-level) algorithm with the two solvers is implemented. The analysis shows that, in the subsonic regions, compared to the AUSM scheme, the HLL solver introduces an additional term about the difference of the left- and right-side reconstructions. The additional term plays a dissipative role, which makes the HLL solver more dissipative than the AUSM solvers. Numerical results show that the BVD algorithm with HLL is more robust than that with AUSM, while the latter can better simulate smooth extrema, contact discontinuities, and small-scale structures. (4) The dissipation characteristics of several typical Riemann solvers are analyzed based on boundary variation, and a modified BVD algorithm is proposed. In the calculation of the modified boundary variation, the Mach number effect and the characteristic velocity effect introduced in the Riemann solver are considered, and a new algorithm is implemented by minimizing this variation. Numerical examples show that the new algorithm can effectively suppress the non-physical overshoot phenomenon that may occur in the original BVD schemes. (5) Based on the local characteristic fields decomposition of compressible Euler equations, a hybrid reconstruction (HR) method is proposed. Toward the system of compressible Euler equations, its local characteristic fields can be decomposed into two genuinely nonlinear fields and the rest linearly degenerate field(s). For genuinely nonlinear fields, it is advisable to use a traditional WENO scheme with better robustness. For linearly degenerate field(s), it is recommended to adopt a higher-resolution scheme such as novel TENO or PnTm-BVD scheme. Compared with traditional WENO methods, the HR method greatly enhances their resolutions for contact discontinuities and smooth extrema. Compared with both TENO and PnTm-BVD schemes, the HR method effectively suppresses their numerical oscillations and spurious overshoots near shocks, and their computational efficiency is improved. (6) Uniformly third- and fourth-order two-stage BVD algorithms are proposed. The PmTn-BVD algorithm has been extended to a third-order P2T2-BVD scheme. Compared with the high-order BVD scheme, the low-order scheme has better stability and robustness. But similar to the third-order WENO scheme, the P2T2-BVD scheme only has second-order accuracy at the second-order critical point. Therefore, in the framework of P4T2-BVD scheme, the three-point second-order polynomial (P2) and fourpoint third-order polynomial (P3) reconstruction are used to replace the P4 polynomial reconstruction in the first-stage BVD selection, and the U3-BVD (uniformly third-order accuracy at the first- and second-order critical points) and U4-BVD (uniformly fourthorder accuracy at the critical points of first three orders) schemes are developed. Compared with the P4T2-BVD, the upwind-biased U3-BVD has better stability, while the non-dissipative U4-BVD has higher resolution for high-frequency waves. (7) The third-order TENO and TENO-T2-BVD schemes are constructed. In the 3rd-order TENO scheme, the hyperbolic secant is suggested to instead the sharp cutoff function. Applying the hyperbolic secant can effectively increase the contribution of sub-smooth templates, thereby the resolution of third-order TENO is improved. Due to the fact that the third-order TENO scheme uses the same stencil as the THINC reconstruction, the TENO-T2-BVD scheme has better robustness. The high-accuracy and high-resolution characteristics of the developed algorithm are verified through numerical examples. (8) A three-stage fourth-order diagonal implicit Runge-Kutta (DIRK) scheme is developed. The novel method has a larger stability region than traditional three-stage third-order explicit RK schemes, and can solve stiff problems, and possesses higher precision than existing three-stage fourth-order DIRK schemes. Combined with the boundary variation diminishing algorithm for spatial discretization, this DIRK method is applied to solve the linear convection equation, nonlinear Burgers equation, and Euler equations. Numerical results show that the proposed DIRK method can obtain stable and accurate solutions, even with relatively larger Courant-Friedrichs-Lewy (CFL) numbers.