Lax-Wendroff 型方法是一种时空耦合的数值方法, 在可压缩流体的数值计算中具有重要意义. 相比时空分离的方法, 这类方法充分利用控制方程进行时空对换, 在数值通量中体现出流场蕴含的多维效应、热力学效应以及其它物理与几何效应, 从而在间断处数值耗散更小, 对流场中的小尺度结构分辨率更高. 本文使用 Lax-Wendroff 型解法器, 围绕可压缩流数值计算的若干问题完成了一系列工作, 包括基于 Lax-Wendroff 型解法器的两步四阶时间离散方法以及与之匹配的紧致型空间重构; 与两步四阶格式匹配的双曲方程边界条件的高精度数值离散; 时空二阶的二维移动边界追踪方法. 一、基于 Lax-Wendroff 型解法器的两步四阶格式 多步龙格-库塔 (R-K) 型方法实现简便, 但存在依赖模板较大的问题. 单步 Lax-Wendroff 型方法依赖模板小, 对间断与小尺度结构分辨率高, 但高阶 Lax-Wendroff 型解法器构造困难、实现复杂. 针对以上两种方法的优势与缺陷, 本文提出了基于 Lax-Wendroff 型解法器的两步四阶时间离散方法以及与之匹配的紧致型空间重构. (1) 两步四阶方法基于二阶 Lax-Wendroff 型解法器, 使用两步计算即可达到四阶时间精度, 与此相对照的 R-K 型方法需要四步计算达到同样的精度. 该方法相比 R-K 方法减少了一半的重构从而节约了计算量, 也使格式更加紧致, 在间断处数值耗散更小, 对涡团等小尺度结构分辨率更高. (2) Lax-Wendroff 型解法器充分利用控制方程, 使得相应的数值格式能够精细刻画流场的多维效应、热力学效应以及其它物理与几何效应. (3) 该方法无需构造单步高阶 Lax-Wendroff 型解法器, 避免了张量计算, 显著简化了数值通量的构造. (4) 与之匹配的空间重构利用 Lax-Wendroff 型解法器给出的解的界面值, 通过高斯定理将其转换为解在计算网格内的梯度. 无需求解额外的非物理方程, 即可应用厄米特型空间重构, 进一步提高了计算格式的紧致性. 二、与两步四阶格式相匹配的双曲方程边界条件的高精度数值离散 作为两步四阶格式的必要完善, 本文构造了与之相匹配的双曲方程边界条件的高精度数值离散. (1) 该方法利用控制方程, 将边界条件给出的解的时间导数转化为空间导数. 从而在边界附近进行插值, 给出计算区域外若干虚拟网格内解的取值, 并在其中蕴含了边界条件. 与以往工作不同是, 该方法仅进行一阶导数的时空互换, 避免了以往方法中的张量计算. (2) 该方法被应用于一维与二维欧拉方程组的固壁边界条件以及拟一维管道流的亚音速出入流等具体的边界条件, 实现了相应算法, 并通过一系列数值实验检验了其有效性. (3) 以往的研究指出多步型数值格式的中间步直接应用精确边界条件会造成格式降阶. 对两步四阶格式的误差分析给出了边界条件修正方法, 结论是边界条件应当与计算区域内部的离散格式相容才能保证格式的整体精度. 三、时空二阶精度移动边界追踪方法 针对刚性固体与流体相互作用的流固耦合问题, 本文构造了二维的时空二阶精度移动边界追踪方法. (1) 设计了单边广义黎曼问题 (单边 GRP) 解法器, 它利用计算区域内的单侧流体状态, 以及移动边界的瞬时速度与加速度, 计算移动边界上的二阶精度数值通量. 相比以往计算格式插值得到的通量, 由单边 GRP 解法器构造的数值通量能够直接体现移动边界上的质量、动量、能量交换. (2) 给出了形状不规则的边界控制体内的线性重构, 关键思想是应用高斯定理. 解的梯度由其界面值得到, 后者由计算区域内部的标准 GRP 解法器以及移动边界处的单边 GRP 解法器给出.

Lax-Wendroff type schemes are of great importance in numerical computations of compressible fluid flows for their time-space coupling property. Compared with time-space decoupled methods, they explicitly utilize governing partial differential equations (PDEs) to perform the time-space coherent procedure to construct numerical fluxes in order to reflect the multi-dimensional, thermodynamical, other physical and geometric effects in the fluid field. The resulting schemes are less dissipative at discontinuities and can resolve better small-scale structures. Using Lax-Wendroff type flow solvers, the present thesis focuses on issues arising from computations of compressible fluid flows and makes several contributions, including the two-stage fourth-order time-accurate discretization based on Lax-Wendroff type flow solvers with the corresponding compact spatial reconstruction, the high-order numerical boundary treatment for hyperbolic PDEs suitable for the two-stage fourth-order scheme, and the two-dimensional second-order space-time accurate moving boundary tracking method. I. Two-stage fourth-order scheme based on Lax-Wendroff type flow solvers. Multi-stage Runge-Kutta (R-K) methods are easy to implement but face the problem of the enlargement of their effective stencils. On the other hand, single-stage Lax-Wendroff methods have advantages of small-size effective stencils and the resulting high resolution at discontinuities and small-scale structures. But high-order Lax-Wendroff type flow solvers are complicated to design and difficult to implement. By taking account of advantages and disadvantages of these two methods, a two-stage fourth-order time-accurate discretization based on Lax-Wendroff type flow solvers is proposed together with the corresponding compact spatial reconstruction. (i) The present approach is based on second-order Lax-Wendroff type flow solvers and can achieve fourth-order accuracy in time by a two-stage procedure, rather than the four-stage procedure used by R-K strategies. Half of the reconstructions are saved. As a result, the numerical scheme is less time-consuming, more compact, less dissipative near discontinuities, and can resolve better small-scale structures such as vortices. (ii) %The use of Lax-Wendroff type flow solvers makes the resulting scheme capture well multi-dimensional, thermodynamical and other physical and geometric effects in the fluid field. Lax-Wendroff type flow solvers can fully utilize the governing PDEs and make the resulting numerical schemes reflect well multi-dimensional, thermodynamical, other physical and geometric effects in the fluid field. (iii) Tensor calculations are avoided by abandoning higher order Lax-Wendroff type flow solvers. The construction of numerical fluxes is dramatically simplified. (iv) In the corresponding spatial reconstruction, the solution gradient is approximated using interface values, in virtue of the Gauss theorem and the advantage of Lax-Wendroff type solvers. As a result, the Hermite-type reconstruction can be applied to further improve the compactness of the scheme without efforts of solving extra non-physical equations. II. High-order numerical boundary treatment for hyperbolic PDEs suitable for the two-stage fourth-order scheme. A high-order numerical boundary treatment for hyperbolic PDEs is developed as the necessary complement for the two-stage fourth-order scheme. (i) The temporal derivative of solution at the boundary is converted to the spatial one using governing PDEs. Interpolation near the boundary is performed and boundary conditions are incorporated into the data in several ghost cells lying outside of the computational domain. Different from previous contributions, the current approach stops at the conversion of first-order derivatives in order to avoid tensor calculations. (ii) The present approach is implemented for several specific boundary conditions including the solid wall boundary condition for the one- and two-dimensional Euler equations, the subsonic inflow and outflow boundary conditions for the quasi one-dimensional nozzle flow. A series of numerical experiments are done to illustrate the performance of the present approach. (iii) As observed in literature, the direct use of exact boundary conditions at intermediate stages causes loss of the accuracy order in multi-stage schemes. The proper modification of boundary conditions is figured out using delicate error analysis for the two-stage fourth-order scheme. We conclude that boundary conditions should be modified consistently with the interior discretization in order to guarantee the accuracy order of the scheme. III. Second-order space-time accurate moving boundary tracking method. A two-dimensional second-order space-time accurate moving boundary tracking method is developed specifically to simulate the interaction between a rigid body and compressible fluids. (i) The one-sided generalized Riemann problem (one-sided GRP) solver is proposed to construct second-order accurate numerical fluxes at moving boundaries. The technique is to use the one-sided fluid state inside the computational domain together with the instantaneous velocity and acceleration of the moving boundary. The resulting numerical fluxes can directly reflect the mass, momentum and energy exchanges through the moving boundary compared with those obtained by interpolations in previous works. (ii) With the idea of using the Gauss theorem, the linear reconstruction for each boundary control volume with an arbitrary shape is developed. The gradient of solution is constructed from its interface values, which are obtained using classical GRP solvers inside the computational domain and the one-sided GRP solver at moving boundaries.