Computational electrodynamics in material media with constraint-preservation, multidimensional Riemann solvers and sub-cell resolution – Part II, higher order FVTD schemes

Balsara, Dinshaw S.; Garain, Sudip K.; Taflove, Allen; Montecinos, Gino I.

doi:10.1016/j.jcp.2017.10.013

Cited by 25 publications

(46 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is again obvious that for a c → ∞ we obtain ε i jk ∂ j J k → B i , i.e. the involution (15). However, the FO-CCZ4 system considered in the rest of this paper will only have homogeneous curl involutions, i.e.…”

Section: Hyperbolic Curl Cleaning With An Extended Generalized Lagranmentioning

confidence: 97%

“…Nowadays, exactly divergence-preserving discontinuous Galerkin and finite volume schemes are available at all orders on Cartesian grids, on structured curvilinear meshes, on unstructured simplex meshes and on geodesic meshes, see e.g. [17,5,13,11,7,10,9,12,15,16,14,49,65]. However, much less is known so far on the construction of arbitrary high order accurate exactly divergence-preserving schemes on general polygonal and polyhedral meshes [67,41,26], or on general space-trees with arbitrary refinement factor r, see e.g.…”

Section: Hyperbolic Curl Cleaning With An Extended Generalized Lagranmentioning

confidence: 99%

See 1 more Smart Citation

On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations

Dumbser

Fambri

Gaburro

et al. 2020

Journal of Computational Physics

View full text Add to dashboard Cite

In this paper we propose an extension of the generalized Lagrangian multiplier method (GLM) of Munz et al. [52,30], which was originally conceived for the numerical solution of the Maxwell and MHD equations with divergence-type involutions, to the case of hyperbolic PDE systems with curl-type involutions.The key idea here is to solve an augmented PDE system, in which curl errors propagate away via a Maxwelltype evolution system. The new approach is first presented on a simple model problem, in order to explain the basic ideas. Subsequently, we apply it to a strongly hyperbolic first order reduction of the CCZ4 formulation (FO-CCZ4) of the Einstein field equations of general relativity, which is endowed with 11 curl constraints. Several numerical examples, including the long-time evolution of a stable neutron star in anti-Cowling approximation, are presented in order to show the obtained improvements with respect to the standard formulation without special treatment of the curl involution constraints.The main advantages of the proposed GLM approach are its complete independence of the underlying numerical scheme and grid topology and its easy implementation into existing computer codes. However, this flexibility comes at the price of needing to add for each curl involution one additional 3 vector plus another scalar in the augmented system for homogeneous curl constraints, and even two additional scalars for non-homogeneous curl involutions. For the FO-CCZ4 system with 11 homogeneous curl involutions, this means that additional 44 evolution quantities need to be added.Keywords: generalized Lagrangian multiplier approach (GLM), hyperbolic PDE systems with curl involutions, Einstein field equations with matter source terms, first order reduction of the CCZ4 system (FO-CCZ4), stable neutron star in anti-Cowling approximation

show abstract

Section: Hyperbolic Curl Cleaning With An Extended Generalized Lagranmentioning

confidence: 97%

Section: Hyperbolic Curl Cleaning With An Extended Generalized Lagranmentioning

confidence: 99%

On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations

Dumbser

Fambri

Gaburro

et al. 2020

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…, , , , , with 1,..., and 1,..., These are the basis functions that we will use in the next section to formulate our ADER-CG scheme. (As an aside, it is also worth mentioning that an ADER-DG scheme would use the same spatial serendipity basis while using temporal basis from Balsara et al 2018; that being the primary difference between ADER-DG and the ADER-CG documented here. )…”

Section: Vb) Serendipity Basis For Spherical Meshes and Efficient Prmentioning

confidence: 99%

“…(An analogous plan was implemented for WENO-AO reconstruction in Balsara et al 2018 resulting in dramatic speed-ups for the WENO algorithm on unstructured meshes and also geodesic meshes; please see Fig. 3 and Section IV of that paper.)…”

Section: Vb) Serendipity Basis For Spherical Meshes and Efficient Prmentioning

confidence: 99%

See 1 more Smart Citation

Efficient, divergence-free, high-order MHD on 3D spherical meshes with optimal geodesic meshing

Balsara

Florinski

Garain

et al. 2019

Monthly Notices of the Royal Astronomical Society

Self Cite

View full text Add to dashboard Cite

There is a great need in several areas of astrophysics and space-physics to carry out high order of accuracy, divergence-free MHD simulations on spherical meshes. This requires us to pay careful attention to the interplay between mesh quality and numerical algorithms. Methods have been designed that fundamentally integrate high order isoparametric mappings with the other high accuracy algorithms that are needed for divergence-free MHD simulations on geodesic meshes.The goal of this paper is to document such algorithms that are implemented in the geodesic mesh version of the RIEMANN code. The fluid variables are reconstructed using a special kind of WENO-AO algorithm that integrates the mesh geometry into the reconstruction process from the ground-up. A novel divergence-free reconstruction strategy for the magnetic field that performs efficiently at all orders, even on isoparametrically mapped meshes, is then presented. The MHD equations are evolved in space and time using a novel ADER predictor algorithm that is efficiently adapted to the isoparametrically mapped geometry. The application of one-dimensional and multidimensional Riemann solvers at suitable locations on the mesh then provides the corrector step. The corrector step for the magnetic field uses a Yee-type staggering of magnetic fields. This results in a scheme with divergence-free update for the magnetic field. The use of ADER enables a one-step update which only requires one messaging operation per complete timestep. This is very 2 beneficial for parallel processing. Several accuracy tests are presented as are stringent test problems. PetaScale performance is also demonstrated on the largest available supercomputers.

show abstract

A Global Divergence Conforming DG Method for Hyperbolic Conservation Laws with Divergence Constraint

Chandrashekar

2018

J Sci Comput

View full text Add to dashboard Cite

We propose a globally divergence conforming discontinuous Galerkin (DG) method on Cartesian meshes for curl-type hyperbolic conservation laws based on directly evolving the face and cell moments of the Raviart-Thomas approximation polynomials. The face moments are evolved using a 1-D discontinuous Gakerkin method that uses 1-D and multi-dimensional Riemann solvers while the cell moments are evolved using a standard 2-D DG scheme that uses 1-D Riemann solvers. The scheme can be implemented in a local manner without the need to solve a global mass matrix which makes it a truly DG method and hence useful for explicit time stepping schemes for hyperbolic problems. The scheme is also shown to exactly preserve the divergence of the vector field at the discrete level. Numerical results using second and third order schemes for induction equation are presented to demonstrate the stability, accuracy and divergence preservation property of the scheme.

show abstract

Computational electrodynamics in material media with constraint-preservation, multidimensional Riemann solvers and sub-cell resolution – Part II, higher order FVTD schemes

Cited by 25 publications

References 48 publications

On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations

On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations

Efficient, divergence-free, high-order MHD on 3D spherical meshes with optimal geodesic meshing

A Global Divergence Conforming DG Method for Hyperbolic Conservation Laws with Divergence Constraint

Contact Info

Product

Resources

About