Multi-dimensional third-order time-implicit scheme for conservation laws

Abstract

When dealing with stiff conservation laws, explicit time integration forces to employ very small time steps, due to the restrictive CFL stability condition. Implicit methods offer an alternative, yielding the possibility to choose the time step according to accuracy constraints. However, the construction of high-order implicit methods is difficult, mainly because of the non-linearity of the space and time limiting procedures required to control spurious oscillations. The Quinpi approach addresses this problem by introducing a first-order implicit predictor, which is employed in both space and time limiting. The scheme has been proposed in (Puppo et al., Comm. Comput. Phys., 2024) for systems of conservation laws in one dimension. In this work the multi-dimensional extension is presented. Similarly to the one-dimensional case, the scheme combines a third-order Central WENO-Z reconstruction in space with a third-order Diagonally Implicit Runge-Kutta (DIRK) method for time integration, and a low order predictor to ease the computation of the Runge-Kutta stages. Even applying space-limiting, spurious oscillations may still appear in implicit integration, especially for large time steps. For this reason, a time-limiting procedure inspired by the MOOD technique and based on numerical entropy production together with a cascade of schemes of decreasing order is applied. The scheme is tested on the Euler equations of gasdynamics also in low Mach regimes. The numerical tests are performed on both structured and unstructured meshes.

Figures (7)

• Figure 1: Radial Sod problem. First row: density at time $t=0.2$ with a grid of $400\times400$ cells and order of accuracy of the solution. The white line in the first panel represents the direction along which the density in the following panels is plotted. Second row: CFL per time step and density profile along the diagonal direction with zoom on the contact and the shock wave. The blue line represents the solution with the Quinpi scheme without time-limiting and the green line the solution with time-limiting and threshold $\gamma=0.05$.
• Figure 2: Rarefaction-contact-shock radial problem. First row: density at time $t=0.75$ and a grid of $400\times400$ cells, order of accuracy of the solution and CFL per time step. The white lines in the first panel represent the direction along which the density in the following panels is plotted. Second row: density profile along the diagonal of the domain and zoom on the discontinuities. Third row: density profile along a radius close to the $x$-axis and zoom on the discontinuities. The red line represents the solution computed with ERK, the blue line the one computed with non-limited-in-time Quinpi and the green line the limited-in-time solution with $\gamma=0.001$.
• Figure 3: Rarefaction-contact-shock radial problem with smaller jump. First row: density at time $t=0.75$ and a grid of $400\times400$ cells and CFL per time step. Second row: profile of the density along the diagonal of the domain and zoom on the contact and shock wave. The limited-in-time Quinpi solution is computed with $\gamma=0.001$.
• Figure 4: Contact-acoustic radial interaction problem: density at time $t=0.35$ with a grid of $400\times400$ cells, CFL per time step and profile of the density along the diagonal in the first quarter of the domain. The limited-in-time Quinpi solution is computed with $\gamma=0.1$.
• Figure 5: Converging-diverging nozzle: density and pressure at time $t=20$ and profile of density, pressure and velocity near the bottom of the domain. The cells in which the solution has been limited in time are marked in black. The white line in the first panels represents the direction along which the solution in the last panel is plotted.

Theorems & Definitions (1)

• Remark 1