Novel High Order Accurate Finite Difference Schemes Constructed via Superconvergence of Finite Element Methods

对新型计算工具的研究不仅可以加深对先进仿真技术的数学理解，而且可以进一步发展现有的流行科学计算软件，从而有利于计算数学和跨学科计算学科的发展。新型有限差分格式的研究将为波传播和对流扩散等物理现象的数值模拟的简化实现提供严格的合理性和稳健性保证，并将在气体动力学、等离子体动力学、惯性约束聚变等领域得到广泛的应用。在众多流行的数值方法中，有限元方法以其丰富的分析理论和对复杂几何问题的灵活性成为最成功的数值方法。另一方面，许多现实世界的应用都是在矩形区域上给出的，或者可以变换到矩形区域上，在矩形区域上，有限差分(FD)型格式因其简单的数据结构和易于实现而成为首选。PI建议探索基于有限元超收敛的新型有限差分格式的构造，以获得更简单的高阶精度数值格式的构造，例如，仅用二次多项式就可以构造四阶精度的有限差分格式。这种方法的一个主要优点是高阶格式中的代数更简单，因为涉及到低阶多项式。此外，简单的代数表示法使分析高阶格式的离散属性变得更容易，例如变系数扩散算子的离散最大值原理。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The research on novel computational tools will benefit both computational mathematics and interdisciplinary computational disciplines with not only deeper mathematical understanding of advanced simulation technology but also further development of existing popular scientific computing software. The study on novel finite difference schemes will provide rigorous justification and robustness guarantee on simplified implementation of high order accurate numerical methods for simulating physical phenomenon such as wave propagation and convection diffusion process with wide applications including gas dynamics, plasma dynamics, inertial confinement fusion, etc. Progress in novel and efficient high order accurate methods will impact on simulation technology in such applications. Among other popular numerical methods, the finite element method has been the most successful one thanks to its rich analysis theories and flexibility for complex geometries. On the other hand, many real world applications are given on or can be transformed to a rectangular domain, on which finite difference (FD) type schemes are preferred due to their simple data structure and easy implementation. The PI proposes to explore construction of novel finite difference type schemes based on superconvergence of finite element method to obtain simpler construction of high order accurate numerical schemes, e.g., a fourth order accurate finite difference scheme can be constructed by using only quadratic polynomials. One major advantage of this approach is the simpler algebra in high order schemes since lower order polynomials are involved. Moreover, simple algebraic representation makes it easier to analyze discrete properties of high order schemes, such as the discrete maximum principle for variable coefficient diffusion operators.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.