Dual-pairing summation-by-parts framework for accurate and efficient numerical simulation of waves and nonlinear hyperbolic conservation laws

Kenneth Duru

Mathematics Sciences Institute, Austrialian National University

The success of modern finite difference (FD) methods for numerical simulation of complex problems in computational mechanics is attributable to the development of summation-by-parts (SBP) finite difference. Traditionally, the design of SBP operators have been exclusively based on central FD stencils on co-located grids, as this has generally been accepted as necessary to ensure a skew-symmetric linear operator which is critical to prove linear stability. Recently, the dual-pairing (DP) SBP framework has shown that this is not necessarily true. The DP SBP operators are a dual-pair of backward and forward FD stencils which together preserve the SBP property. Because of the additional degrees of freedom, the DP SBP framework supports the design of SBP FD with improved properties, such as upwinding and dispersion relations preserving (DRP) properties, necessary for reliable simulation of nonlinear problems, including shocks, and wave propagation problems with high-frequency components. The result of this improvement is the absence of computationally fatal spurious wave modes in numerically computed solutions, and an efficiency increase that is exponential with the dimension of the problem. We will define and give explicit examples of DP SBP operators with a complete methodology to construct them.We will present numerical simulations of complex wave problems in 3D elastic solids and nonlinear atmospheric fluid flow, and demonstrate the efficiency of the DP SBP framework over traditional methods.

Biography

Dr Kenneth Duru is a computational and applied mathematician. He is currently a Fellow at the Mathematical Sciences Institute, Australian National University. In 2012, he earned his PhD in Scientific Computing (major Numerical Analysis) from Uppsala University, Sweden. Dr Duru’s research lie at the interfaces of mathematical analysis, numerical analysis and high-performance computing (HPC), and contributes to the mathematical foundation of numerical methods and simulation tools for the solution of partial differential equations modelling complex real-world problems.