Addressing the challenges of implementation of high-order finite volume schemes for atmospheric dynamics of unstructured meshes

The solution of the non-hydrostatic compressible Euler equations using Weighted Essentially Non-Oscillatory (WENO) schemes in two and three-dimensional unstructured meshes, is presented. Their key characteristics are their simplicity; accuracy; robustness; non-oscillatory properties; versatility in handling any type of grid topology; computational and parallel efficiency. Their defining characteristic is a non-linear combination of a series of high-order reconstruction polynomials arising from a series of reconstruction stencils. In the present study an explicit TVD Runge-Kutta 3rd -order method is employed due to its lower computational resources requirement compared to implicit type time advancement methods. The WENO schemes (up to 5th -order) are applied to the two dimensional and three dimensional test cases: a 2D rising

Implementation of a low-mach number modification for high-order finite-volume schemes for arbitrary hybrid unstructured meshes

An implementation of a novel low-mach number treatment for high-order finite-volume schemes using arbitrary hybrid unstructured meshes is presented in this paper. Low-Mach order modifications for Godunov type finite-volume schemes have been implemented successfully for structured and unstructured meshes, however the methods break down for hybrid mesh topologies containing multiple element types. The modification is applied to the UCNS3D finite-volume framework for compressible flow configurations, which have been shown as very capable of handling any type of grid topology. The numerical methods under consideration are the Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and the Weighted Essentially Non-Oscillatory (WENO) schemes for two-dimensional mixed-element type unstructured meshes. In the present study the HLLC Approximate Riemann Solver is used with an explicit TVD Runge-Kutta 3rd-order method due to its excellent scalability. These schemes (up to 5th-order) are applied to well established two-dimensional and three-dimensional test cases. The challenges that occur when applying these methods to low-mach flow configurations is thoroughly analysed and possible improvements and further test cases are suggested

Space-time adaptive solution of inverse problems with the discrete adjoint method

Adaptivity in both space and time has become the norm for solving problems modeled by partial differential equations. The size of the discretized problem makes uniformly refined grids computationally prohibitive. Adaptive refinement of meshes and time steps allows to capture the phenomena of interest while keeping the cost of a simulation tractable on the current hardware. Many fields in science and engineering require the solution of inverse problems where parameters for a given model are estimated based on available measurement information. In contrast to forward (regular) simulations, inverse problems have not extensively benefited from the adaptive solver technology. Previous research in inverse problems has focused mainly on the continuous approach to calculate sensitivities, and has typically employed fixed time and space meshes in the solution process. Inverse problem solvers that make exclusive use of uniform or static meshes avoid complications such as the differentiation of mesh motion equations, or inconsistencies in the sensitivity equations between subdomains with different refinement levels. However, this comes at the cost of low computational efficiency. More efficient computations are possible through judicious use of adaptive mesh refinement, adaptive time steps, and the discrete adjoint method. This paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent, adaptive grid, adaptive step models. Discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation. However, this approach brings several important challenges. The adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations. A reduction in accuracy of the discrete adjoint sensitivities may appear due to the intergrid transfer operators. Moreover, the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations. This work shows that several of these potential issues can be avoided for the discontinuous Galerkin (DG) method. The adjoint model development is considerably simplified by decoupling the adaptive mesh refinement mechanism from the forward model solver, and by selectively applying automatic differentiation on individual algorithms. In forward models discontinuous Galerkin discretizations can efficiently handle high orders of accuracy, $h/p$-refinement, and parallel computation. The analysis reveals that this approach, paired with Runge Kutta time stepping, is well suited for the adaptive solutions of inverse problems. The usefulness of discrete discontinuous Galerkin adjoints is illustrated on a two-dimensional adaptive data assimilation problem

WENO schemes on unstructured meshes using a relaxed a posteriori MOOD limiting approach

In this paper a relaxed formulation of the a posteriori Multi-dimensional Optimal Order Detection (MOOD) limiting approach is introduced for weighted essentially non-oscillatory (WENO) finite volume schemes on unstructured meshes. The main goal is to minimise the computational footprint of the MOOD limiting approach by employing WENO schemes—by virtue of requiring a smaller number of cells to reduce their order of accuracy compared to an unlimited scheme. The key characteristic of the present relaxed MOOD formulation is that the Numerical Admissible Detector (NAD) is not uniquely defined for all orders of spatial accuracy, and it is relaxed when reaching a 2nd-order of accuracy. The augmented numerical schemes are applied to the 2D unsteady Euler equations for a multitude of test problems including the 2D vortex evolution, cylindrical explosion, double-Mach reflection, and an implosion. It is observed that in many events, the implemented MOOD paradigm manages to preserve symmetry of the forming structures in simulations, an improvement comparing to the non-MOOD limited counterparts which cannot be easily obtained due to the multi-dimensional reconstruction nature of the schemes

Extended bounds limiter for high-order finite-volume schemes on unstructured meshes

This paper explores the impact of the definition of the bounds of the limiter proposed by Michalak and Ollivier-Gooch in [Accuracy preserving limiter for the high-order accurate solution of the Euler equations, J. Comput. Phys. 228 (2009) 8693–8711], for higher-order Monotone-Upstream Central Scheme for Conservation Laws (MUSCL) numerical schemes on unstructured meshes in the finite-volume (FV) framework. A new modification of the limiter is proposed where the bounds are redefined by utilising all the spatial information provided by all the elements in the reconstruction stencil. Numerical results obtained on smooth and discontinuous test problems of the Euler equations on unstructured meshes, highlight that the newly proposed extended bounds limiter exhibits superior performance in terms of accuracy and mesh sensitivity compared to the cell-based or vertex-based bounds implementations