Algorithm and solver development
Authors: Wustenberg Henrik , Khurana Parv , Xia Boyang , Ye Junjie 
Published: June 2024

First session on the algorithm and solver developments within Nektar++ Framework. The session comprises the following four talks
1. Towards fast and robust incompressible flow simulations with Nektar++
2. Lower-order refined preconditioning for spectral/hp element methods for complex, 3D geometries
3. Progress on an efficient discontinuous Galerkin incompressible Navier-Stokes solver
4. Implementation of the ALE method for implicit compressible solver in Nektar++Towards fast and robust incompressible flow simulations with Nektar++Many Nektar++ users rely on the incompressible Navier-Stokes solver for their research on automotive aerodynamics, wind turbines, plunging wings or other challenging problems. These problems involve highly unsteady flows around complex geometries which challenge the stability of numerical algorithms in reliably predicting the flow physics. A major constraint for stability is the CFL conditions that constraints the maximum stable time step size for high Reynolds number flows. This work presents a linear-implicit time stepping scheme for the incompressible solver and focusses on the possible improvement in stability. Moreover, the limitation to small time steps typically forces a temporal over-resolution. A case study on a Formula 1 geometry shows that we can exploit larger time steps for computational performance on these challenging flow problems.
Lower-order refined preconditioning for spectral/hp element methods for complex, 3D geometriesEfficient preconditioning techniques are needed to reduce the computational cost for the iterative solution of spectral hp/element methods to simulate high Reynolds number incompressible flows around complex, 3D industrial geometries. This work presents a preconditioning technique, Lower-order refined (LOR) preconditioner, for the Poisson problem, typically seen in the decoupled pressure equation system in the incompressible Navier-Stokes equations solver within Nektar++.

LOR preconditioner, also known as the SEMFEM preconditioner, uses a spectrally equivalent lower-order (P=1) discretization to precondition the high-order problem. The LOR preconditioner exhibits a bounded iterative condition number with increasing problem size, making it desirable for large-scale problems. It utilizes lower-order operator sparsity for cheaper operator evaluations and a constant memory requirement per degree of freedom. It shows minimal sensitivity to high aspect-ratio elements, making it suitable for complex geometries. This work addresses the previous gaps, as the LOR preconditioner is demonstrated for spectral/hp elements in Nektar++, using a modal expansion basis for triangle elements in 2D and prismatic and tetrahedron elements in 3D - in addition to quadrilaterals in 2D and hexahedrons in 3D. The solver performance of the LOR preconditioner is analysed in practice using suitable test cases of increasing complexity relevant to the context of race-car aerodynamics.
Progress on an efficient discontinuous Galerkin incompressible Navier-Stokes solverThe ever-increasing floating-point performance of modern hardware enables a promising future in computational fluid dynamics. To exploit such performance, mitigating memory-bound effects and getting a new tradeoff is vital, which has led to significant developments around matrix-free designs. The discontinuous Galerkin method (DG) is most suitable for matrix-free but so far Nektar++ does not have a robust and efficient implementation. In this talk, we will present the current progress towards an efficient discontinuous Galerkin incompressible Navier-Stokes solver, with a focus on the design and optimization of the implicit part - the Helmholtz solver. For the algorithm, we will discuss some key points and unresolved issues which affect the convergence and accuracy of the multigrid interior penalty method. And for the optimization, we will present a new efficient matrix-free DG framework, which is built upon new inter-element mappings so that we can coalesce as many operators as possible to reduce memory pressure.Implementation of the ALE method for implicit compressible solver in Nektar++This presentation will describe the implementation of the implicit compressible sliding mesh solver within the Nektar++ framework. This project is inspired by the need to perform high-fidelity turbomachinery simulations, particularly in analyzing the impact of incoming wakes. Compared to the explicit solve, the implicit solver provides unconditional stability, allowing for bigger time steps and enhancing computational efficiency. The presentation will offer a concise introduction and delineate the methodology behind constructing the implicit sliding mesh solver. Additionally, the architecture and several test results of the implicit sliding mesh solver will be displayed.