# Space-time adaptive ADER-DG schemes for dissipative flows: Compressible Navier-Stokes and resistive MHD equations

@article{Fambri2016SpacetimeAA, title={Space-time adaptive ADER-DG schemes for dissipative flows: Compressible Navier-Stokes and resistive MHD equations}, author={Francesco Fambri and Michael Dumbser and Olindo Zanotti}, journal={Comput. Phys. Commun.}, year={2016}, volume={220}, pages={297-318} }

## 49 Citations

### Discontinuous Galerkin Methods for Compressible and Incompressible Flows on Space–Time Adaptive Meshes: Toward a Novel Family of Efficient Numerical Methods for Fluid Dynamics

- Computer Science
- 2020

Two new families of spectral semi-implicit and spectral space–time DG methods for the solution of the two and three dimensional Navier–Stokes equations on edge-based adaptive staggered Cartesian grids are derived.

### Discontinuous Galerkin Methods for Compressible and Incompressible Flows on Space–Time Adaptive Meshes: Toward a Novel Family of Efficient Numerical Methods for Fluid Dynamics

- Computer ScienceArchives of Computational Methods in Engineering
- 2019

The presented results show clearly that the high-resolution and shock-capturing capabilities of the news schemes are significantly enhanced within the cell-by-cell AMR implementation together with time accurate local time stepping (LTS).

### An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: Theory and numerical verification

- Computer ScienceJ. Comput. Phys.
- 2020

### Discontinuous Galerkin methods for compressible and incompressible flows on space-time adaptive meshes

- Computer Science
- 2017

In this work the numerical discretization of the partial differential governing equations for compressible and incompressible flows is dealt within the discontinuous
Galerkin (DG) framework along…

### Efficient implementation of space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell WENO finite-volume limiting for simulation of non-stationary compressible multicomponent reactive flows

- MathematicsJournal of Scientific Computing
- 2023

The present work is devoted to the study of efficient implementation of spacetime adaptive ADER finite element discontinuous Galerkin method with a posteriori correction technique of solutions on…

### Semi-implicit discontinuous Galerkin methods for the incompressible Navier–Stokes equations on adaptive staggered Cartesian grids

- Computer Science
- 2016

### ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics

- Computer Science
- 2018

A new class of high-order accurate numerical algorithms for solving the equations of general-relativistic ideal magnetohydrodynamics in curved spacetimes is presented, which is as robust as classical second-order total-variation diminishing finite-volume methods at shocks and discontinuities, but also as accurate as unlimited high- order DG schemes in smooth regions of the flow.

### On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

- MathematicsJournal of Scientific Computing
- 2021

A thermodynamically compatible viscous extension of the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamic compatible numerical scheme is considered.

### High Order ADER Schemes for Continuum Mechanics

- Computer ScienceFrontiers in Physics
- 2020

The unified symmetric hyperbolic and thermodynamically compatible (SHTC) formulation of continuum mechanics developed by Godunov, Peshkov, and Romenski is presented, which allows to describe fluid and solid mechanics in one single and unified first orderhyperbolic system.

### Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine

- Computer ScienceAxioms
- 2018

These are the largest runs ever carried out with high order ADER-DG schemes for nonlinear hyperbolic PDE systems and a detailed performance comparison with traditional Runge-Kutta DG schemes is provided.

## 126 References

### A Discontinuous Galerkin Scheme based on a Space-Time Expansion II. Viscous Flow Equations in Multi Dimensions

- MathematicsJ. Sci. Comput.
- 2008

In numerical simulations for the two-dimensional compressible Navier-Stokes equations, the efficiency and the optimal order of convergence being p+1, when a polynomial approximation of degree p is used.

### Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

- Computer Science
- 2015

### High-Order Unstructured One-Step PNPM Schemes for the Viscous and Resistive MHD Equations

- Computer Science
- 2009

The new, unified framework of high order onestep PNPM schemes recently proposed for inviscid hyperbolic conservation laws is used in order to solve the viscous and resistive magnetohydrodynamics equations in two and three space dimensions on unstructured triangular and tetrahedral meshes.

### Semi-implicit discontinuous Galerkin methods for the incompressible Navier–Stokes equations on adaptive staggered Cartesian grids

- Computer Science
- 2016

### Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case

- Computer Science, Mathematics
- 2005

### Spectral semi-implicit and space-time discontinuous Galerkin methods for the incompressible Navier-Stokes equations on staggered Cartesian grids

- Computer Science
- 2016

### High order space–time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems

- Computer Science
- 2014

### Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems

- Computer ScienceJ. Sci. Comput.
- 2001

The theoretical and algorithmic aspects of the Runge–Kutta discontinuous Galerkin methods are reviewed and several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations are shown.

### A staggered space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations on two-dimensional triangular meshes

- Computer Science
- 2014

### Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes

- Computer ScienceJ. Comput. Phys.
- 2013