629 research outputs found
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
An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs
We extend the entropy stable high order nodal discontinuous Galerkin spectral
element approximation for the non-linear two dimensional shallow water
equations presented by Wintermeyer et al. [N. Wintermeyer, A. R. Winters, G. J.
Gassner, and D. A. Kopriva. An entropy stable nodal discontinuous Galerkin
method for the two dimensional shallow water equations on unstructured
curvilinear meshes with discontinuous bathymetry. Journal of Computational
Physics, 340:200-242, 2017] with a shock capturing technique and a positivity
preservation capability to handle dry areas. The scheme preserves the entropy
inequality, is well-balanced and works on unstructured, possibly curved,
quadrilateral meshes. For the shock capturing, we introduce an artificial
viscosity to the equations and prove that the numerical scheme remains entropy
stable. We add a positivity preserving limiter to guarantee non-negative water
heights as long as the mean water height is non-negative. We prove that
non-negative mean water heights are guaranteed under a certain additional time
step restriction for the entropy stable numerical interface flux. We implement
the method on GPU architectures using the abstract language OCCA, a unified
approach to multi-threading languages. We show that the entropy stable scheme
is well suited to GPUs as the necessary extra calculations do not negatively
impact the runtime up to reasonably high polynomial degrees (around ). We
provide numerical examples that challenge the shock capturing and positivity
properties of our scheme to verify our theoretical findings
Embedded discontinuous Galerkin transport schemes with localised limiters
Motivated by finite element spaces used for representation of temperature in
the compatible finite element approach for numerical weather prediction, we
introduce locally bounded transport schemes for (partially-)continuous finite
element spaces. The underlying high-order transport scheme is constructed by
injecting the partially-continuous field into an embedding discontinuous finite
element space, applying a stable upwind discontinuous Galerkin (DG) scheme, and
projecting back into the partially-continuous space; we call this an embedded
DG scheme. We prove that this scheme is stable in L2 provided that the
underlying upwind DG scheme is. We then provide a framework for applying
limiters for embedded DG transport schemes. Standard DG limiters are applied
during the underlying DG scheme. We introduce a new localised form of
element-based flux-correction which we apply to limiting the projection back
into the partially-continuous space, so that the whole transport scheme is
bounded. We provide details in the specific case of tensor-product finite
element spaces on wedge elements that are discontinuous P1/Q1 in the horizontal
and continuous P2 in the vertical. The framework is illustrated with numerical
tests
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
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