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

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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.