用特征线法和样条有限元法求解奇异映射盘状区域上的双曲椭圆问题
Solving hyperbolic-elliptic problems on singular mapped disk-like domains with the method of characteristics and spline finite elements
摘要：
在偏微分方程的数值求解中，一个常见的策略是定义一个张量积多维逻辑域的统一离散化，通过给定的坐标变换将其映射到一个物理域。通过将这一概念扩展到多块设置，可以在相对复杂的几何形状上采用简单而有效的数值算法。这种方法的主要缺点是处理坐标变换的奇异性的固有困难。这项工作提出了一个全面的数值策略的常见情况下的盘状域的奇点在一个独特的极点，其中一个边缘的矩形逻辑域崩溃的物理域（例如，一个圆圈）。我们提出了强大的数值方法的解决方案Vlasov双曲型方程耦合到Poisson椭圆型方程在这样的几何形状。我们描述了一种半拉格朗日平流求解器，该求解器采用一组新的坐标，称为伪笛卡尔坐标，以在整个域中集成特征方程，包括极点，以及基于全局$C^1$光滑样条的有限元椭圆求解器（Toshniwal等人，2017年）。这两个求解器进行测试，独立和耦合模型，即磁化等离子体的2D导向中心模型，相当于不可压缩的无粘欧拉流体的涡度模型。所提出的数值方法在空间离散化参数中显示出高阶收敛性，均匀地跨越计算域，而没有由于奇异性而导致的降阶效果。专门的测试表明，所描述的数值技术也可以直接应用于点电荷（相当于，点状漩涡）的存在下，在粒子网格法的上下文中。
英文摘要：
A common strategy in the numerical solution of partial differential equations is to define a uniform discretization of a tensor-product multi-dimensional logical domain, which is mapped to a physical domain through a given coordinate transformation. By extending this concept to a multi-patch setting, simple and efficient numerical algorithms can be employed on relatively complex geometries. The main drawback of such an approach is the inherent difficulty in dealing with singularities of the coordinate transformation. This work suggests a comprehensive numerical strategy for the common situation of disk-like domains with a singularity at a unique pole, where one edge of the rectangular logical domain collapses to one point of the physical domain (for example, a circle). We present robust numerical methods for the solution of Vlasov-like hyperbolic equations coupled to Poisson-like elliptic equations in such geometries. We describe a semi-Lagrangian advection solver that employs a novel set of coordinates, named pseudo-Cartesian coordinates, to integrate the characteristic equations in the whole domain, including the pole, and a finite element elliptic solver based on globally $C^1$ smooth splines (Toshniwal et al., 2017). The two solvers are tested both independently and on a coupled model, namely the 2D guiding-center model for magnetized plasmas, equivalent to a vorticity model for incompressible inviscid Euler fluids. The numerical methods presented show high-order convergence in the space discretization parameters, uniformly across the computational domain, without effects of order reduction due to the singularity. Dedicated tests show that the numerical techniques described can be applied straightforwardly also in the presence of point charges (equivalently, point-like vortices), within the context of particle-in-cell methods.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
