# Single-Step Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Method for 1-D Euler Equations

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## Abstract

We propose an explicit, single-step discontinuous Galerkin method on moving grids using the arbitrary Lagrangian–Eulerian approach for one-dimensional Euler equations. The grid is moved with the local fluid velocity modified by some smoothing, which is found to considerably reduce the numerical dissipation introduced by Riemann solvers. The scheme preserves constant states for any mesh motion and we also study its positivity preservation property. Local grid refinement and coarsening are performed to maintain the mesh quality and avoid the appearance of very small or large cells. Second, higher order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme.

## Keywords

Discontinuous Galerkin method Moving meshes Arbitrary Lagrangian–Eulerian Euler equations## Mathematics Subject Classification

65M60 35L04## 1 Introduction

Finite volume schemes based on exact or approximate Riemann solvers are used for solving hyperbolic conservation laws like the Euler equations governing compressible flows. These schemes are able to compute discontinuous solutions in a stable manner since they have implicit dissipation built into them due to the upwind nature of the schemes. Higher order schemes are constructed following a reconstruction approach combined with a high order time integration scheme. Discontinuous Galerkin methods can be considered as higher order generalizations of finite volume methods which also make use of Riemann solver technology but do not need a reconstruction step since they evolve a polynomial solution inside each cell. While these methods are formally high order accurate on smooth solutions, they can still introduce too much numerical dissipation in some situations. Springel [1] gives the example of a Kelvin–Helmholtz instability in which adding a large constant velocity to both states leads to suppression of the instability due to excessive numerical dissipation. This behaviour is attributed to the fact that fixed grid methods based on upwind schemes are not Galilean invariant. Upwind schemes, even when they are formally high order accurate, are found to be too dissipative when applied to turbulent flows [2] since the numerical viscosity can overwhelm the physical viscosity.

*a*| which is the wave speed. In case of Euler equations simulated with a Riemann solver, e.g., the Roe scheme, the wave speeds are related to the eigenvalues of the flux Jacobian and the numerical dissipation would be proportional to the absolute values of the eigenvalues, e.g., \(|v-c|, |v|, |v+c|\) where

*v*is the fluid velocity and

*c*is the sound speed. This type of the numerical viscosity is not Galilean invariant since the fluid velocity depends on the coordinate frame adopted for the description of the flow. Adding a large translational velocity to the coordinate frame will increase the numerical viscosity and reduce the accuracy of the numerical solution. Such high numerical viscosity can be eliminated or minimized if the grid moves along with the flow as in Lagrangian methods [3, 4, 5]. However, pure Lagrangian methods encounter the issue of large grid deformations that occur in highly sheared flows as in the Kelvin–Helmholtz problem requiring some form of re-meshing. A related approach is to use the arbitrary Lagrangian–Eulerian approach [6, 7] where the mesh velocity can be chosen to be close to the local fluid velocity but may be regularized to maintain the mesh quality. Even in the ALE approach, it may be necessary to perform some local remeshing to prevent the grid quality from degrading. In [1], the mesh is regenerated after every time step based on a Delaunay triangulation, which allows it to maintain good mesh quality even when the fluid undergoes large shear deformation. However, these methods have been restricted to second-order accuracy as they rely on unstructured finite volume schemes on general polygonal/polyhedral cells, where achieving higher order accuracy is much more difficult compared to structured grids.

Traditionally, ALE methods have been used for problems involving moving boundaries as in wing flutter, store separation and other problems involving fluid structure interaction [8, 9, 10, 11, 12]. In these applications, the main reason to use ALE is not to minimize the dissipation in upwind schemes but to account for the moving boundaries and, hence, the grid velocities are chosen based on boundary motion and with a view to maintain good mesh quality. Another class of methods solves the PDE on moving meshes where the mesh motion is determined based on a monitor function which is designed to detect regions of large gradients in the solution, see [13, 14] and the references therein. These methods achieve automatic clustering of grid points in regions of large gradients. ALE schemes have been used to compute multi-material flows as in [15], since they are useful to accurately track the material interface. The mesh velocity was chosen to be equal to the contact speed but away from the material contact, the velocity was chosen by linear interpolation and was not close to Lagrangian. There are other methods for choosing the mesh velocity which have been studied in [16, 17]. Lax–Wendroff type ALE schemes for compressible flows have been developed in [18]. Finite volume schemes based on the ADER approach have been developed on unstructured grids [19, 20, 21]. The theoretical analysis of ALE-DG schemes in the framework of Runge–Kutta time stepping for conservation laws has been done in [22].

In the present work, we consider only the one-dimensional problem to set down the fundamental principles with which in an upcoming work, we shall solve the multi-dimensional problem. The numerical method developed here will be usable in the multiple dimensions, but additional work is required in multiple dimensions to maintain a good mesh quality under fluid flow deformation. We develop an explicit discontinuous Galerkin scheme that is conservative on moving meshes and automatically satisfies the geometric conservation law. The scheme is a single-step method which is achieved using a predictor computed from a Runge–Kutta scheme that is local to each cell in the sense that it does not require any data from neighbouring cells and belongs to the class of schemes called the ADER method. Due to the single-step nature of the scheme, the TVD limiter has to be applied only once in each time step unlike in multi-stage Runge–Kutta schemes where the limiter is applied after each stage update. This nature of the ADER scheme can reduce its computational expense, especially in multi-dimensional problems and while performing parallel computations. The mesh velocity is specified at each cell face as the local velocity with some smoothing. We analyze the positivity of the first-order scheme using the Rusanov flux and derive a CFL condition. The scheme is shown to be exact for steady moving contact waves and the solutions are invariant to the motion of the coordinate frame. Due to the Lagrangian nature, the Roe scheme does not require any entropy fix. However, we identify the possibility of spurious contact waves arising in some situations. This is due to the vanishing of the eigenvalue corresponding to the contact wave. While the cell averages are well predicted, the higher moments of the solution can be inaccurate. This behaviour of Lagrangian DG schemes does not seem to have been reported in the literature. We propose a fix for the eigenvalue in the spirit of the entropy fix of Harten [23] that prevents the spurious contact waves from occurring in the solution. The methodology developed here will be extended to multi-dimensional flows in a future work with a view towards handling complex sheared flows.

The rest of the paper is organized as follows. Section 2 introduces the Euler equation model that is used in the rest of the paper. In Sect. 3, we explain the derivation of the scheme on a moving mesh together with the quadrature approximations and computation of mesh velocity. The computation of the predicted solution is detailed in Sect. 4. The TVD type limiter is presented in Sect. 5 for a non-uniform mesh, Sect. 6 shows the positivity of the first-order scheme and Sect. 7 shows the preservation of constant states. The grid coarsening and refinement strategy are explained in Sect. 8, while Sect. 9 presents a series of numerical results.

## 2 Euler Equations

*v*is the fluid velocity,

*p*is the pressure and

*E*is the total energy per unit volume, which for an ideal gas is given by \(E = p/(\gamma -1) + \rho v^2/2\), with \(\gamma > 1\) being the ratio of specific heats at constant pressure and volume, and \(H=(E+p)/\rho\) is the enthalpy. The Euler equations form a hyperbolic system; the flux Jacobian \(A(\varvec{u}) = \varvec{f}'(\varvec{u})\) has real eigenvalues and linearly independent eigenvectors. The eigenvalues are \(v-c, \ v, \ v+c\) where \(c=\sqrt{\gamma p/\rho }\) is the speed of sound and the corresponding right eigenvectors are given by

*A*can be diagonalized as \(A = R \varLambda R^{-1}\) where

*R*is the matrix formed by the right eigenvectors as the columns and \(\varLambda\) is the diagonal matrix of eigenvalues.

## 3 Discontinuous Galerkin Method

### 3.1 Mesh and Solution Space

*j*th cell being denoted by \(C_j(t) = [x_{j - \frac{1}{2}}(t), x_{j + \frac{1}{2}}(t)]\). As the notation shows, the cell boundaries are time dependent which means that the cell is moving in some specified manner. The time levels are denoted by \(t_n\) with the time step \(\Delta t_n = t_{n+1} - t_n\). The boundaries of the cells move with a constant velocity in the time interval \((t_n, t_{n+1})\) given by

*w*(

*x*,

*t*) be the continuous linear interpolation of the mesh velocity which is given by

*j*th cell is given by

*j*th cell. The basis functions \(\varphi _m\) are defined in terms of Legendre polynomials

*m*. The above definition of the basis functions implies the following orthogonality property:

*j*th cell in terms of the reference coordinates \(\xi\) as

### 3.2 Derivation of the Scheme

*l*th moment of the solution starting from

*w*(

*x*,

*t*) is linear in

*x*and hence \(\frac{\partial w}{\partial x}\) is constant inside each cell. Hence, the

*l*th moment evolves according to

*x*variable, we obtain

### 3.3 Mesh Velocity

- i.The first choice is to take an average of the two velocities at every face. In the numerical results, we refer to this as ADG$$\begin{aligned} {\tilde{w}}_{j + \frac{1}{2}}^n = \frac{1}{2}\bigg [ v\bigg (x_{j + \frac{1}{2}}^-,t_n\bigg ) + v\bigg (x_{j + \frac{1}{2}}^+, t_n\bigg )\bigg ]. \end{aligned}$$
- ii.The second choice is to solve a linearized Riemann problem at the face at time \(t_n\). In the numerical results, we refer to this as RDG. For simplicity of notation, let the solution to the left of the face \(x_{j+\frac{1}{2}}\) be represented as \(u_{j+\frac{1}{2}}^-\) and the solution to the right be represented as \(u_{j+\frac{1}{2}}^+\). Then,$$\begin{aligned} {\tilde{w}}_{j + \frac{1}{2}}^n = \frac{\rho _j^nc_j^nv_j^n + \rho _{j+1}^nc_{j+1}^nv_{j+1}^n}{\rho _j^nc_j^n + \rho _{j+1}^nc_{j+1}^n} + \frac{p_j^n - p_{j+1}^n}{\rho _j^nc_j^n + \rho _{j+1}^nc_{j+1}^n}. \end{aligned}$$

### Remark 1

## 4 Computing the Predictor

*t*and

*x*is sufficient. Consider a quadrature point \((x_q,\tau _r)\), the Taylor expansion of the solution around the cell center \(x_j^n\) and time level \(t_n\) yields

*k*. These nodes are moving with velocity

*w*(

*x*,

*t*), so that the time evolution of the solution at node \(x_m\) is governed by

- i.
compute nodal values \(\varvec{U}_m(\tau _r)\), \(m=0,1,\cdots ,k;\)

- ii.
for each

*r*, convert the nodal values to modal coefficients \(\varvec{u}_{m,r}\), \(m=0,1,\cdots ,k;\) - iii.
evaluate predictor \(\varvec{U}_h(x_q,\tau _r) = \sum \limits _{m=0}^k \varvec{u}_{m,r} \varphi _m(x_q, \tau _r).\)

### Remark 2

By looping over each cell in the mesh, the predicted solution is computed in each cell and the cell integral in (6) is evaluated. The trace values \(\varvec{U}_{j + \frac{1}{2}}^-(\tau _r)\) and \(\varvec{U}_{j - \frac{1}{2}}^+(\tau _r)\) at the \(\Box\) points needed for quadrature in time are computed and stored. These are later used in a loop over the cell faces where the numerical flux is evaluated. Thus, the algorithm is easily parallelizable on multiple core machines and/or using threads.

## 5 Control of Oscillation by Limiting

High order schemes for hyperbolic equations suffer from spurious numerical oscillations when discontinuities or large gradients are present in the solution which cannot be accurately resolved on the mesh. In the case of scalar problems, this is a manifestation of loss of TVD property and hence limiters are used to satisfy some form of TVD condition. In the case of DG schemes, the limiter is used as a post-processor which is applied on the solution after the time update has been performed. If the limiter detects that the solution is oscillatory, then the solution polynomial is reduced to at most a linear polynomial with a limited slope. In the present scheme, the limiter is applied after the solution is updated from time \(t_n\) to time \(t_{n+1}\), i.e., the solution \(\varvec{u}_h^{n+1}\) obtained from (6) is post-processed by the limiter. Since the mesh is inherently non-uniform due to it being moved with the flow, we modify the standard TVD limiter to account for this non-uniformity. Also, since we are solving a system of conservation laws, the limiter is applied on the local characteristic variables which gives better results than applying it directly on the conserved variables [26].

*j*can be written as

^{1}

*M*is an estimate of the second derivative of the solution at smooth extrema [27] and has to be chosen by the user.

## 6 Positivity Property

^{2}

### Theorem 1

*The scheme* (9) *with the Rusanov flux is positivity preserving if the time step condition* (10) *is satisfied*.

### Remark 3

In this work, we have not attempted to prove the positivity of the scheme for other numerical fluxes. We also do not have a proof of positivity for higher order version of the scheme. In the computations, we use the positivity preserving limiter of [28] which leads to robust schemes which preserve the positivity of the cell average value in all the test cases.

## 7 Preservation of Constant States

*w*is an affine function of

*x*and \(\varphi _l\) are orthogonal. This implies that \(\varvec{u}_{j,l}^{n+1}=0\) for \(l=1,2,\cdots ,k\). For \(l=0\), we get

## 8 Grid Coarsening and Refinement

The size of the cells can change considerably during the time evolution process due to the near Lagrangian movement of the cell boundaries. Near shocks, the cells will be compressed to smaller sizes which will reduce the allowable time step since a CFL condition has to be satisfied. In some regions, e.g., inside expansion fans, the cell size can increase considerably which may lead to loss of accuracy. To avoid too small or too large cells from occurring in the grid, we implement cell merging and refinement into our scheme. If a cell becomes smaller than some specified size \(h_{\min }\), then it is merged with one of its neighbouring cells and the solution is transferred from the two cells to the new cell by performing an \(L^2\) projection. If a cell size becomes larger than some specified size \(h_{\max }\), then this cell is refined into two cells by division and the solution is again transferred by the \(L^2\) projection. The use of the \(L^2\) projection for solution transfer ensures the conservation of mass, momentum and energy and preserves the accuracy in smooth regions. We also ensure that the cell sizes do not change drastically between neighbouring cells. To keep a track of refinement of cells, each cell is assigned an initial level equal to 0. The daughter cells created during refinement are assigned a level incremented from the parent cell, while the coarsened cells are assigned a level decremented from the parent cell.

The algorithm for refinement and coarsening is carried out in three sweeps over all the active cells. In the first sweep, we mark the cells for refinement or coarsening based on their size and the level of neighboring cells. Cells are marked for coarsening if the size is less than a pre-specified minimum size. They are marked for refinement if either the size of the cell is larger than the maximum size or if the level of the cell is less than the level of the neighboring cells. If none of the conditions are satisfied, the cells are marked for no change. In the second sweep, a cell is marked for refinement if both the neighboring cells are marked for refinement. A cell is also marked for refinement if the size of the cell is larger than twice the size of either of the neighboring cells, and is also larger than twice the minimum size. The last condition is inserted to prevent a cell being alternately marked for refinement and coarsening in consecutive adaptation cycles. In the third and final sweeps, we again mark cells for refinement if both the neighboring cells are marked for refinement. Further, we ensure that a cell marked for refinement does not have a neighboring cell marked for a coarsening, since this can lead to an inconsistent mesh.

## 9 Numerical Results

- i.
choose the mesh velocity \(w_{j + \frac{1}{2}};\)

- ii.
choose the time step \(\Delta t_n;\)

- iii.
compute the predictor \(\varvec{U}_h;\)

- iv.
update the solution \(\varvec{u}_h^n\) to the next time level \(\varvec{u}_h^{n+1};\)

- v.
apply the TVD/TVB limiter on \(\varvec{u}_h^{n+1};\)

- vi.
apply the positivity limiter on \(\varvec{u}_h^{n+1}\) from [28];

- vii.
perform grid refinement/coarsening.

### 9.1 Order of Accuracy

Order of accuracy study on static mesh using Rusanov flux

| \(k=1\) | \(k=2\) | \(k=3\) | |||
---|---|---|---|---|---|---|

Error | Rate | Error | Rate | Error | Rate | |

100 | 4.370E−02 | – | 3.498E−03 | – | 3.883E−04 | – |

200 | 6.611E−03 | 2.725 | 4.766E−04 | 2.876 | 1.620E−05 | 4.583 |

400 | 1.332E−03 | 2.518 | 6.415E−05 | 2.885 | 9.376E−07 | 4.347 |

800 | 3.151E−04 | 2.372 | 8.246E−06 | 2.910 | 5.763E−08 | 4.239 |

1 600 | 7.846E−05 | 2.280 | 1.031E−06 | 2.932 | 3.595E−09 | 4.180 |

Order of accuracy study on moving mesh using Rusanov flux

| \(k=1\) | \(k=2\) | \(k=3\) | |||
---|---|---|---|---|---|---|

Error | Rate | Error | Rate | Error | Rate | |

100 | 2.331E−02 | – | 3.979E−03 | – | 8.633E−04 | – |

200 | 6.139E−03 | 1.9250 | 4.058E−04 | 3.294 | 1.185E−05 | 6.186 |

400 | 1.406E−03 | 2.0258 | 5.250E−05 | 3.122 | 7.079E−07 | 5.126 |

800 | 3.375E−04 | 2.0366 | 6.626E−06 | 3.077 | 4.340E−08 | 4.760 |

1 600 | 8.278E−05 | 2.0344 | 8.304E−07 | 3.057 | 2.689E−09 | 4.573 |

Order of accuracy study on static mesh using HLLC flux

| \(k=1\) | \(k=2\) | \(k=3\) | |||
---|---|---|---|---|---|---|

Error | Rate | Error | Rate | Error | Rate | |

100 | 4.582E−02 | – | 3.952E−03 | – | 3.464E−04 | – |

200 | 9.611E−03 | 2.253 | 4.048E−04 | 3.287 | 2.058E−05 | 4.073 |

400 | 2.052E−03 | 2.240 | 4.640E−05 | 3.206 | 1.287E−06 | 4.036 |

800 | 4.803E−04 | 2.192 | 5.623E−06 | 3.152 | 8.061E−08 | 4.023 |

1 600 | 1.184E−04 | 2.149 | 6.929E−07 | 3.119 | 5.050E−09 | 4.016 |

Order of accuracy study on moving mesh using HLLC flux

| \(k=1\) | \(k=2\) | \(k=3\) | |||
---|---|---|---|---|---|---|

Error | Rate | Error | Rate | Error | Rate | |

100 | 1.590E−02 | – | 1.626E−03 | – | 1.962E−04 | – |

200 | 4.042E−03 | 1.977 | 2.072E−04 | 2.972 | 1.269E−05 | 3.950 |

400 | 1.014E−03 | 1.985 | 2.605E−05 | 2.982 | 7.983E−07 | 3.971 |

800 | 2.538E−04 | 1.990 | 3.261E−06 | 2.988 | 4.997E−08 | 3.980 |

1 600 | 6.349E−05 | 1.992 | 4.077E−07 | 2.991 | 3.124E−09 | 3.985 |

Order of accuracy study on moving mesh using Rusanov flux using higher order limiter [29]

| \(k=1\) | \(k=2\) | ||
---|---|---|---|---|

Error | Rate | Error | Rate | |

100 | 2.053E−02 | – | 2.277E−03 | – |

200 | 4.312E−03 | 2.251 | 3.425E−04 | 2.732 |

400 | 1.031E−03 | 2.064 | 4.565E−05 | 2.907 |

800 | 2.550E−04 | 2.015 | 5.812E−06 | 2.973 |

1 600 | 6.356E−05 | 2.004 | 7.315E−07 | 2.990 |

Order of accuracy study on moving mesh using HLLC flux with randomly perturbed mesh velocity

| \(k=1\) | \(k=2\) | \(k=3\) | |||
---|---|---|---|---|---|---|

Error | Rate | Error | Rate | Error | Rate | |

100 | 1.735E−02 | – | 1.798E−03 | – | 2.351E−04 | – |

200 | 4.179E−03 | 2.051 | 2.848E−04 | 2.676 | 1.416E−05 | 4.069 |

400 | 1.054E−03 | 2.035 | 4.301E−05 | 2.703 | 8.578E−07 | 4.041 |

800 | 2.615E−04 | 1.943 | 6.012E−06 | 2.838 | 5.476E−08 | 3.958 |

1 600 | 7.279E−05 | 1.852 | 8.000E−07 | 2.909 | 3.505E−09 | 3.966 |

### 9.2 Smooth Test Case with Non-constant Velocity

Order of accuracy study on fixed mesh using Roe flux with non-constant velocity smooth test case

| \(k=1\) | \(k=2\) | ||
---|---|---|---|---|

Error | Rate | Error | Rate | |

100 | 8.535E−03 | – | 1.033E−03 | – |

200 | 1.958E−03 | 2.124 | 1.221E−04 | 3.08 |

400 | 4.721E−04 | 2.052 | 1.581E−05 | 2.95 |

800 | 1.238E−04 | 1.931 | 2.14E−06 | 2.89 |

1 600 | 3.563E−05 | 1.796 | 2.63E−07 | 3.02 |

Order of accuracy study on moving mesh using Roe flux with non-constant velocity smooth test case

| \(k=1\) | \(k=2\) | ||
---|---|---|---|---|

Error | Rate | Error | Rate | |

100 | 4.235E−03 | – | 2.238E−04 | – |

200 | 1.058E−03 | 2.001 | 3.255E−05 | 2.87 |

400 | 2.586E−04 | 2.035 | 4.301E−05 | 3.133 |

800 | 5.804E−05 | 2.155 | 5.762E−06 | 2.901 |

1 600 | 1.271E−05 | 2.192 | 7.401E−07 | 2.96 |

### 9.3 Single Contact Wave

### 9.4 Sod Problem

Number of iterations required to reach time \(t=0.2\) for Sod test for different boost velocity of the coordinate frame

| 0 | 10 | 100 |
---|---|---|---|

Static mesh | 144 | 810 | 6 807 |

Moving mesh | 176 | 176 | 176 |

### 9.5 Lax Problem

### 9.6 Shu–Osher Problem

### 9.7 Titarev–Toro Problem

### 9.8 123 Problem

### 9.9 Blast Problem

### 9.10 Le Blanc Shock Tube Test Case

### 9.11 Two-Dimensional Isentropic Vortex Test Case

Isentropic vortex in 2D: order of accuracy study on two-dimensional static mesh

| \(k=1\) | \(k=2\) | ||
---|---|---|---|---|

Error | Rate | Error | Rate | |

50 \(\times\) 50 | 2.230E−03 | – | 1.762E−04 | – |

100 \(\times\) 100 | 5.987E−04 | 1.945 | 2.305E−05 | 2.934 |

200 \(\times\) 200 | 1.498E−04 | 1.998 | 2.973E−06 | 2.955 |

400 \(\times\) 400 | 3.786E−05 | 1.984 | 3.762E−07 | 2.982 |

800 \(\times\) 800 | 9.617E−06 | 1.977 | 3.474E−08 | 2.991 |

Isentropic vortex in 2D: order of accuracy study on two-dimensional moving mesh

| \(k=1\) | \(k=2\) | ||
---|---|---|---|---|

Error | Rate | Error | Rate | |

50 \(\times\) 50 | 2.230E−03 | – | 1.762E–04 | – |

100 \(\times\) 100 | 5.987E−04 | 1.945 | 2.305E−05 | 2.934 |

200 \(\times\) 200 | 1.498E−04 | 1.998 | 2.973E−06 | 2.955 |

400 \(\times\) 400 | 3.786E−05 | 1.984 | 3.762E−07 | 2.982 |

800 \(\times\) 800 | 9.617E−06 | 1.977 | 3.474E−08 | 2.991 |

## 10 Summary and Conclusions

We have developed an explicit DG scheme on moving meshes using ALE framework and space–time expansion of the solutions within each cell. The near Lagrangian nature of the mesh motion dramatically reduces the numerical dissipation, especially for contact waves. Even moving contact waves can be exactly computed with a numerical flux that is exact for stationary contact waves. The scheme is shown to yield superior results even in the presence of the large boost velocity of the coordinate system indicating its Galilean invariance property. The standard Roe flux does not suffer from entropy violation when applied in the current nearly Lagrangian framework. However, in some problems with strong shocks, spurious contact waves can appear and we propose to fix the dissipation in Roe-type schemes that eliminates this issue. The method yields accurate solutions even in combination with standard TVD limiters, where fixed grid methods perform poorly. The mesh motion provides automatic grid adaptation near shocks but may lead to very coarse cells inside expansion waves. A grid adaptation strategy is developed to handle the problem of very small or very large cells. The presence of the DG polynomials makes it easy to transfer the solution during grid adaptation without loss of accuracy. The proposed methodology is general enough to be applicable to other systems of conservation laws modelling fluid flows. The basic idea can be extended to multi-dimensions but additional considerations are required to maintain good mesh quality under fluid deformations. The preliminary results shown for the isentropic vortex are very promising for the 2-D case.

## Footnotes

## Notes

### Acknowledgements

This work was started when Jayesh Badwaik was a project assistant at TIFR-CAM, Bangalore. The first two authors gratefully acknowledge the financial support received from the Airbus Foundation Chair on Mathematics of Complex Systems established in TIFR-CAM, Bangalore, for carrying out this work. Christian Klingenberg acknowledges the support of the *Priority Program 1648*: *Software for Exascale Computing* by the German Science Foundation. On behalf of all authors, the corresponding author states that there is no conflict of interest.

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