Well-Balanced Nodal Discontinuous Galerkin Method for Euler Equations with Gravity
Abstract
We present a well-balanced nodal discontinuous Galerkin (DG) scheme for compressible Euler equations with gravity. The DG scheme makes use of discontinuous Lagrange basis functions supported at Gauss–Lobatto–Legendre (GLL) nodes together with GLL quadrature using the same nodes. The well-balanced property is achieved by a specific form of source term discretization that depends on the nature of the hydrostatic solution, together with the GLL nodes for quadrature of the source term. The scheme is able to preserve isothermal and polytropic stationary solutions upto machine precision on any mesh composed of quadrilateral cells and for any gravitational potential. It is applied on several examples to demonstrate its well-balanced property and the improved resolution of small perturbations around the stationary solution.
Keywords
Discontinuous Galerkin Euler equations Gravity Well-balanced1 Introduction
The Euler equations in the presence of a gravitational field are an important mathematical model arising in atmospheric flows and astrophysical applications. Due to the presence of gravitational force, these equations have non-trivial stationary solutions, usually refered to as hydrostatic solutions. These stationary solutions are of interest in themselves and particularly their stability to small perturbations. Many atmospheric phenoma are small perturbations around the hydrostatic solution. The accurate computation of these small perturbations about the hydrostatic solution is hence important in many applications. In general, there are many other mathematical models involving source terms which exhibit non-trivial stationary solutions. An important class of such models with source terms are the shallow water equations arising in river and ocean modeling. Any numerical scheme which preserves the hydrostatic solution on any mesh is said to be well-balanced.
Stationary solutions are obtained by the precise balance of fluxes and source terms. Schemes which are not well-balanced will not be able to achieve this precise balance and may give rise to large numerical errors close to the stationary solutions, especially on coarse meshes [4, 20]. In order to obtain reliable solutions, such schemes would require very fine meshes, which may be impractical in realistic simulations in three dimensions. Well-balanced schemes on the other hand yield accurate solutions even on coarse meshes and are capable of resolving small perturbations around the stationary solution.
Li and Xing [14] have proposed a well-balanced DG scheme using orthogonal basis functions for Euler equations with gravity for isothermal hydrostatic solutions. The well-balanced property is achieved by a re-writing of the source terms together with an integration by parts, and using a Lax–Friedrich type numerical flux with a modified viscosity. Since orthogonal basis functions are used, the initial condition is projected onto the finite element space.
In this work, we propose a well-balanced DG scheme for isothermal and polytropic hydrostatic solutions under the ideal gas assumption. The scheme is based on nodal Lagrange basis functions using Gauss–Lobatto–Legendre points on arbitrary quadrilateral cells in 2-D. The same GLL points are also used for quadrature in the weak formulation of the DG scheme. The source term is re-written based on whether we are near isothermal or polytropic hydrostatic solution and then discretized using the GLL points. For continuous isothermal and polytropic hydrostatic solutions, the scheme is well-balanced for any consistent numerical flux function. The scheme is also well-balanced for isothermal hydrostatic solutions in which density might be discontinuous provided we use a numerical flux function which is exact for stationary contact discontinuities, like the Roe or HLLC flux, and the initial discontinuity in density coincides with the cell boundaries. For discontinuous solutions, a non-linear TVD limiter is necessary to avoid unphysical oscillations. The limiter might destroy the well-balanced property but this is easily solved by preventing the application of the limiter in case the solution residual in any cell is zero (close to machine precision), which would be the case for a hydrostatic solution.
The rest of the paper is organized as follows. In Sect. 2 we introduce the 1-D Euler equations, explain the hydrostatic solutions, introduce the DG scheme and prove its well-balanced property. In Sect. 3 we perform the same steps for the 2-D Euler equations and explain the limiter in Sect. 4. Numerical results are shown in Sect. 5 to demonstrate the well-balanced property and the accurate resolution of perturbations around hydrostatic solutions. Finally we end the paper with a summary and conclusions.
2 1-D Euler Equations with Gravity
2.1 Hydrostatic States
2.1.1 Isothermal Solution
2.1.2 Polytropic Solution
2.2 Mesh and Basis Functions
2.3 Semi-discrete DG Scheme in 1-D
Remark 1
2.4 Numerical Flux Function
2.5 Approximation of Source Term
If the potential is known to us as an explicit function of the spatial coordinate, we can compute the source term exactly by taking its derivative, but this does not lead to a well balanced scheme. In order to achieve the well-balanced property we will find it useful to approximate the spatial derivative of the potential in a form similar to the flux derivative in the above DG scheme.
2.5.1 Isothermal Case
2.5.2 Polytropic Case
Theorem 2.1
Let the initial condition be obtained by interpolating the hydrostatic solution corresponding to a continuous gravitational potential \(\Phi \). Then the DG scheme (2.6) together with the source term approximation given by (2.8) or (2.9) preserves the initial condition under any time integration scheme.
Proof
Remark The use of GLL nodes was important in the above proof. The boundary flux terms vanish since the GLL nodes ensure that the interpolation of the hydrostatic solution on the mesh is continuous across the elements. The entire scheme makes use of only the solution at the GLL nodes which is exact in the hydrostatic case and helps us to satisfy the well-balanced property.
3 2-D Euler Equations with Gravity
3.1 Hydrostatic Solution
3.2 Mesh and Basis Functions
3.3 Semi-discrete DG Scheme
3.4 Approximation of Source Term
If we denote the source terms in the momentum equation as \((s_h^x, s_h^y)\), then the source term in the energy equation is given by \( \frac{1}{\rho _h}[ (\rho u)_h s_h^x + (\rho v)_h s_h^y]\). This completes the specification of the DG scheme.
Theorem 3.1
Let the initial condition be obtained by interpolating the hydrostatic solution corresponding to a continuous gravitational potential \(\Phi \). Then the DG scheme (3.2) together with the source term approximation given by (3.3) or (3.4) preserves the initial condition under any time integration scheme.
Proof
4 Limiter
As described above, the limiter is applied component-wise to the conserved variables. However it is beneficial to apply the limiter to characteristic variables [5]. The average gradient is transformed by multiplying with the matrix of left eigenvectors. The limiter is applied as above and transformed back to the original variables by multiplying with the matrix of right eigenvectors.
The application of limiter can change the solution in a cell even if the solution is hydrostatic and smooth, especially at extrema. In order to preserve the well-balanced property of the scheme, we apply the limiter in a cell only if the \(L^2\) norm of the cell residual is greater than a small number and in the computations we use a tolerance of \(10^{-12}\). In all the numerical tests, we find that this is sufficient to prevent the application of limiter when the solution is in hydrostatic state.
5 Numerical Results
5.1 1-D Hydrostatic Solution
5.1.1 Isothermal Case
Well-balanced test for isothermal case on Cartesian mesh using \(Q_1\) polynomials and potential \(\Phi = x\)
Mesh | \(\rho u\) | \(\rho v\) | \(\rho \) | E |
---|---|---|---|---|
\(25\times 25\) | 1.03822e–13 | 6.68114e–15 | 2.72604e–14 | 9.53913e–14 |
\(50\times 50\) | 1.04783e–13 | 5.92391e–15 | 2.67559e–14 | 9.36725e–14 |
\(100\times 100\) | 1.05019e–13 | 5.6383e–15 | 2.66323e–14 | 9.34503e–14 |
\(200\times 200\) | 1.05088e–13 | 5.54862e–15 | 2.66601e–14 | 9.33861e–14 |
Well-balanced test for isothermal case on Cartesian mesh using \(Q_2\) polynomials and potential \(\Phi =x\)
Mesh | \(\rho u\) | \(\rho v\) | \(\rho \) | E |
---|---|---|---|---|
\(25\times 25\) | 1.04518e–13 | 7.29936e–15 | 2.7548e–14 | 9.64205e–14 |
\(50\times 50\) | 1.04983e–13 | 6.03994e–15 | 2.69317e–14 | 9.43158e–14 |
\(100\times 100\) | 1.05069e–13 | 5.68612e–15 | 2.69998e–14 | 9.39126e–14 |
\(200\times 200\) | 1.05089e–13 | 5.69125e–15 | 2.68828e–14 | 9.462e–14 |
Well-balanced test for isothermal case on Cartesian mesh using \(Q_1\) polynomials and \(\Phi = \sin (2\pi x)\)
Mesh | \(\rho u\) | \(\rho v\) | \(\rho \) | E |
---|---|---|---|---|
\(25\times 25\) | 9.23424e–13 | 1.16432e–13 | 2.31405e–13 | 8.16645e–13 |
\(50\times 50\) | 9.36459e–13 | 1.04921e–13 | 2.28315e–13 | 8.04602e–13 |
\(100\times 100\) | 9.39613e–13 | 1.00384e–13 | 2.28001e–13 | 8.03005e–13 |
\(200\times 200\) | 9.40422e–13 | 9.89098e–14 | 2.2792e–13 | 8.02653e–13 |
Well-balanced test for isothermal case on Cartesian mesh using \(Q_2\) polynomials and \(\Phi = \sin (2\pi x)\)
Mesh | \(\rho u\) | \(\rho v\) | \(\rho \) | E |
---|---|---|---|---|
\(25\times 25\) | 9.34536e–13 | 1.32134e–13 | 2.35173e–13 | 8.30316e–13 |
\(50\times 50\) | 9.39556e–13 | 1.08172e–13 | 2.29908e–13 | 8.10055e–13 |
\(100\times 100\) | 9.40442e–13 | 1.00923e–13 | 2.28538e–13 | 8.04638e–13 |
\(200\times 200\) | 9.40668e–13 | 9.90613e–14 | 2.28051e–13 | 8.0357e–13 |
5.1.2 Polytropic Case
5.2 Order of Accuracy Study When Scheme is Not Well-Balanced
In this test, we start with the polytropic hydrostatic solution from Sect. 5.1.2 as initial condition in two dimensions with gravity acting vertically, and solve it with the isothermal well-balanced scheme. This scheme will not be able to exactly preserve the polytropic solution. Since the exact solution is the polytropic hydrostatic solution, we can compute the \(L^2\) norm of the error. The error is computed on different grid sizes and polynomials degrees, and the results are shown in Tables 7 and 8. The horizontal momentum error is close to machine zero since gravity acts on in the vertical direction. The other quantities are not exactly preserved but their errors converge at the expected rate, i.e., we get second order accuracy with \(Q_1\) polynomials and third order accuracy with \(Q_2\) polynomials.
5.3 2-D Hydrostatic Solution: I
Well-balanced test for polytropic case on Cartesian mesh using \(Q_1\) polynomials and potential \(\Phi = x\)
Mesh | \(\rho u\) | \(\rho v\) | \(\rho \) | E |
---|---|---|---|---|
\(25\times 25\) | 1.07749e–13 | 6.67302e–15 | 2.89086e–14 | 9.38882e–14 |
\(50\times 50\) | 1.08760e–13 | 5.87560e–15 | 2.82644e–14 | 9.17933e–14 |
\(100\times 100\) | 1.07487e–13 | 5.29355e–15 | 4.91244e–14 | 9.62769e–14 |
\(200\times 200\) | 1.09086e–13 | 5.70042e–15 | 2.81423e–14 | 9.14467e–14 |
Well-balanced test for polytropic case on Cartesian mesh using \(Q_2\) polynomials and potential \(\Phi = x\)
Mesh | \(\rho u\) | \(\rho v\) | \(\rho \) | E |
---|---|---|---|---|
\(25\times 25\) | 1.08483e–13 | 7.35525e–15 | 2.92146e–14 | 9.48770e–14 |
\(50\times 50\) | 1.08977e–13 | 6.03037e–15 | 2.85168e–14 | 9.25119e–14 |
\(100\times 100\) | 1.09071e–13 | 5.75922e–15 | 2.86190e–14 | 9.23044e–14 |
\(200\times 200\) | 1.09107e–13 | 6.11341e–15 | 2.91511e–14 | 9.39398e–14 |
5.4 2-D Hydrostatic Solution: II
Case 1 Here we choose \(T_l = 1\), \(T_u = 2\). This corresponds to lighter fluid on top of heavier fluid which is a stable configuration. The DG scheme preserves the initial condition upto machine precision as seen in Table 10.
Polytropic hydrostatic solution solved with isothermal well-balanced scheme on Cartesian mesh using \(Q_1\) polynomials and potential \(\Phi = x\)
Mesh size | \(\rho u\) | \(\rho v\) | \(\rho \) | E | |||
---|---|---|---|---|---|---|---|
Error | Error | Rate | Error | Rate | Error | Rate | |
\(25\times 25\) | 6.22039e–14 | 1.39945e–05 | – | 5.03134e–07 | – | 1.50727e–06 | – |
\(50\times 50\) | 6.27891e–14 | 3.51615e–06 | 1.99 | 1.71697e–07 | 1.55 | 4.03669e–07 | 1.90 |
\(100\times 100\) | 6.29344e–14 | 8.79605e–07 | 1.99 | 4.91080e–08 | 1.80 | 1.08737e–07 | 1.89 |
\(200\times 200\) | 6.29710e–14 | 2.19966e–07 | 1.99 | 1.30477e–08 | 1.91 | 2.83352e–08 | 1.94 |
Polytropic hydrostatic solution solved with isothermal well-balanced scheme on Cartesian mesh using \(Q_2\) polynomials and potential \(\Phi = x\)
Mesh size | \(\rho u\) | \(\rho v\) | \(\rho \) | E | |||
---|---|---|---|---|---|---|---|
Error | Error | Rate | Error | Rate | Error | Rate | |
\(25\times 25\) | 6.26353e–14 | 1.03474e–07 | – | 1.17234e–07 | – | 3.80288e–07 | – |
\(50\times 50\) | 6.24467e–14 | 1.29041e–08 | 3.00 | 1.46356e–08 | 3.00 | 4.74617e–08 | 3.00 |
\(100\times 100\) | 6.29693e–14 | 1.61142e–09 | 3.00 | 1.82873e–09 | 3.00 | 5.92946e–09 | 3.00 |
\(200\times 200\) | 6.29792e–14 | 2.01344e–10 | 3.00 | 2.28559e–10 | 3.00 | 7.41017e–10 | 3.00 |
Well-balanced test for 2-D isothermal hydrostatic solution on Cartesian meshes
\(\rho u\) | \(\rho v\) | \(\rho \) | E | |
---|---|---|---|---|
\(Q_1\), \(25\times 25\) | 9.85926e–14 | 9.85855e–14 | 5.32357e–14 | 1.55361e–13 |
\(Q_1\), \(50\times 50\) | 9.94493e–14 | 9.94451e–14 | 5.37084e–14 | 1.56669e–13 |
\(Q_1\), \(100\times 100\) | 9.96481e–14 | 9.96474e–14 | 5.38404e–14 | 1.57062e–13 |
\(Q_2\), \(25\times 25\) | 9.9256e–14 | 9.92682e–14 | 5.39863e–14 | 1.57435e–13 |
\(Q_2\), \(50\times 50\) | 9.961e–14 | 9.96538e–14 | 5.41091e–14 | 1.57521e–13 |
\(Q_2\), \(100\times 100\) | 9.95889e–14 | 9.97907e–14 | 5.43145e–14 | 1.57728e–13 |
Well-balanced test for Rayleigh–Taylor problem in Case 1, \(T_l=1\), \(T_u=2\)
\(\rho u\) | \(\rho v\) | \(\rho \) | E | |
---|---|---|---|---|
\(Q_1, 25\times 100\) | 5.13108e–13 | 1.10971e–13 | 2.56836e–13 | 8.91784e–13 |
\(Q_1, 50\times 200\) | 5.15744e–13 | 1.12234e–13 | 2.84309e–13 | 8.97389e–13 |
\(Q_2, 25\times 100\) | 5.15726e–13 | 1.1341e–13 | 3.22713e–13 | 9.01183e–13 |
\(Q_2, 50\times 200\) | 5.16397e–13 | 1.13707e–13 | 3.93623e–13 | 9.02503e–13 |
Well-balanced test for Rayleigh–Taylor problem in Case 2, \(T_l=2\), \(T_u=1\)
\(\rho u\) | \(\rho v\) | \(\rho \) | E | |
---|---|---|---|---|
\(Q_1, 25\times 100\) | 3.487e–13 | 1.0989e–13 | 1.98949e–13 | 7.42265e–13 |
\(Q_1, 50\times 200\) | 3.50153e–13 | 1.1121e–13 | 2.34991e–13 | 7.4747e–13 |
\(Q_2, 25\times 100\) | 3.50149e–13 | 1.12159e–13 | 2.80645e–13 | 7.5104e–13 |
\(Q_2, 50\times 200\) | 3.50548e–13 | 1.12533e–13 | 3.63806e–13 | 7.52151e–13 |
5.5 Order of Accuracy Study
Convergence of error for degree \(N=1\)
1 / h | \(\rho u\) | \(\rho v\) | \(\rho \) | E | ||||
---|---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | Error | Rate | |
50 | 0.00134154 | – | 0.00134154 | – | 0.0012837 | – | 0.00161287 | – |
100 | 0.000335446 | 1.99 | 0.000335446 | 1.99 | 0.00032044 | 2.00 | 0.000411141 | 1.97 |
200 | 8.35627e–05 | 2.00 | 8.35627e–05 | 2.00 | 7.97842e–05 | 2.00 | 0.00010335 | 1.99 |
400 | 2.08348e–05 | 2.00 | 2.08348e–05 | 2.00 | 1.98754e–05 | 2.00 | 2.58109e–05 | 2.00 |
Convergence of error for degree \(N=2\)
1 / h | \(\rho u\) | \(\rho v\) | \(\rho \) | E | ||||
---|---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | Error | Rate | |
25 | 7.7019e–05 | – | 7.7019e–05 | – | 7.80868e–05 | – | 9.32865e–05 | – |
50 | 9.68863e–06 | 2.99 | 9.68863e–06 | 2.99 | 9.76471e–06 | 2.99 | 1.16849e–05 | 2.99 |
100 | 1.21506e–06 | 2.99 | 1.21506e–06 | 2.99 | 1.22031e–06 | 3.00 | 1.46256e–06 | 2.99 |
200 | 1.52134e–07 | 2.99 | 1.52134e–07 | 2.99 | 1.52503e–07 | 3.00 | 1.8247e–07 | 3.00 |
5.6 Radial Rayleigh–Taylor Instability
Well balanced test for radial Rayleigh–Taylor problem on Cartesian mesh
\(\rho u\) | \(\rho v\) | \(\rho \) | E | |
---|---|---|---|---|
\(Q_1, 30\times 30\) | 6.08113e–13 | 6.06081e–13 | 2.11611e–14 | 4.13127e–14 |
\(Q_1, 50\times 50\) | 5.45634e–13 | 5.44973e–13 | 1.29319e–14 | 2.66376e–14 |
\(Q_1, 100\times 100\) | 5.4918e–13 | 5.49012e–13 | 9.63433e–15 | 2.11463e–14 |
\(Q_2, 30\times 30\) | 6.29776e–13 | 6.27278e–13 | 1.9777e–14 | 5.08689e–14 |
\(Q_2, 50\times 50\) | 5.5376e–13 | 5.52903e–13 | 1.5635e–14 | 3.85134e–14 |
\(Q_2, 100\times 100\) | 5.51645e–13 | 5.5142e–13 | 2.173e–14 | 5.25578e–14 |
Well balanced test for radial Rayleigh–Taylor problem on unstructured mesh
\(\rho u\) | \(\rho v\) | \(\rho \) | E | |
---|---|---|---|---|
\(Q_1\), 956 | 3.03033e–16 | 3.23738e–16 | 6.66245e–16 | 2.11449e–16 |
\(Q_1\), 2037 | 5.20653e–16 | 5.10865e–16 | 9.82565e–16 | 4.06402e–16 |
\(Q_1\), 10,710 | 2.00037e–15 | 1.57078e–15 | 2.21284e–15 | 4.66515e–15 |
\(Q_2\), 956 | 2.69429e–14 | 3.31591e–14 | 4.50499e–14 | 1.61455e–13 |
\(Q_2\), 2037 | 7.68549e–14 | 1.1834e–13 | 1.01632e–13 | 3.84886e–13 |
\(Q_2\), 10,710 | 2.92633e–13 | 2.44323e–13 | 2.9449e–13 | 4.10069e–13 |
5.7 Shock Tube Problem
6 Summary
We have proposed a DG scheme for Euler equations with gravitational source terms which is well-balanced for isothermal and polytropic hydrostatic solutions. The scheme is based on nodal Lagrange basis functions using Gauss–Lobatto–Legendre points. The well-balanced property holds for arbitrary gravitational potentials and even on unstructured grids. Standard numerical flux functions can be used in this approach. The scheme is able to compute perturbations around the hydrostatic solution accurately even on coarse meshes.
Notes
Acknowledgements
Praveen Chandrashekar thanks the Airbus Foundation Chair for Mathematics of Complex Systems at TIFR-CAM, Bangalore, for support in carrying out this work. Markus Zenk thanks the DAAD Passage to India program which supported his visit to Bangalore during which time part of this work was conducted.
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