Applied Math Colloquium | Siva Nadarajah, Shock Capturing in Nonlinearly Stable High-Order Methods

MC 5501 

Speaker

Siva Nadarajah, Department of Mechanical Engineering, McGill University.

Title

Shock Capturing in Nonlinearly Stable High-Order Methods

Abstract

Solutions to the governing partial differential equations obtained from a discrete numerical scheme can have significant errors, especially near shocks when the discrete representation of the solution cannot fully capture the discontinuity in the solution. Split forms and entropy-conserving flux differencing approaches have become popular since they guarantee robustness for unsteady problems on coarse unstructured grids. Nonlinearly stable flux reconstruction (NSFR) combines the key properties of provable nonlinear stability and the increased time-step from energy-stable flux reconstruction. NSFR has successfully been applied to unsteady compressible flows in arbitrary curvilinear coordinates while utilizing low-storage weight-adjusted approaches to scale efficiently with low memory consumption. Unfortunately, these schemes fail to satisfy the maximum principle for scalar conservation laws nor do they preserve positivity in the case of hyperbolic conservation laws. In this talk, we present our current contributions for treating both stationary and moving shocks.

For moving shocks we incorporate bound-preserving limiters within an NSFR framework to obtain a robust solution that preserves the desired properties while maintaining a high-order of accuracy. As for stationary shocks, we have developed a novel full-space approach that converges the mesh, shape and solution simultaneously. The convergence of the optimization problem is independent of the size of the problem and attained through the use of second-order information while maintaining a tractable computational and implementation cost by employing automatic differentiation. Furthermore, a novel preconditioner based on the reduced-space is extended to accommodate additional constraints to avoid ill-conditioning of the resulting enormous full-space system. In engineering applications, the solution field is often used to evaluate a scalar functional of interest, such as lift or drag over an airfoil. While functionals are sensitive to errors in the flow solution, certain regions in the domain are more important for accurate evaluation of the functional than the rest. Using this fact, we formulate a goal-oriented implicit shock-tracking approach that captures a segment of the shock that is important for evaluating the functional. Shock tracking is achieved using Lagrange-Newton-Krylov-Schur (LNKS) full space optimizer, with the objective of minimizing the adjoint-weighted residual error indicator. The method is applied to test cases of two-dimensional advection and inviscid compressible flows to demonstrate functional-dependent shock tracking. Tracking the entire shock without using artificial dissipation results in the error converging at the orders of O(hp+1).