New Adaptive Numerical Methods Based on Dual Formulation of Hyperbolic Conservation Laws

TL;DR

This work addresses the challenge of achieving high-resolution, efficient simulations for hyperbolic conservation laws by combining dual formulations. By evolving both the conservative and nonconservative forms and using their discrepancies to build a smoothness indicator, the method adaptively selects discretizations that are highly accurate in smooth regions and robust near discontinuities. The approach is demonstrated on the Euler equations in 1-D and 2-D, with region-specific schemes including central-upwind fluxes, A-WENO variants, and overcompressive limiters to sharpen contact and other discontinuities. The results show improved resolution of complex flow features at reduced computational cost compared with non-adaptive fifth-order A-WENO, highlighting the practical potential of the DF-based adaptive framework for complex, multiscale gas-dynamics problems and suggesting directions for future multi-fluid and asymptotic-preserving extensions.

Abstract

In this paper, we propose an adaptive high-order method for hyperbolic systems of conservation laws. The proposed method is based on a dual formulation approach: Two numerical solutions, corresponding to conservative and nonconservative formulations of the same system, are evolved simultaneously. Since nonconservative schemes are known to produce nonphysical weak solutions near discontinuities, we exploit the difference between these two solutions to construct a smoothness indicator (SI). In smooth regions, the difference between the conservative and nonconservative solutions is of the same order as the truncation error of the underlying discretization, whereas in nonsmooth regions, it is ${\cal O}(1)$. We apply this idea to the Euler equations of gas dynamics and define the SI using differences in the momentum and pressure variables. This choice allows us to further distinguish neighborhoods of contact discontinuities from other nonsmooth parts of the computed solution. The resulting classification is used to adaptively select numerical discretizations. In the vicinities of contact discontinuities, we employ the low-dissipation central-upwind numerical flux and a second-order piecewise linear reconstruction with the slopes computed using an overcompressive SBM limiter. Elsewhere, we use an alternative weighted essentially non-oscillatory (A-WENO) framework with the central-upwind finite-volume numerical fluxes and either unlimited (in smooth regions) or Ai-WENO-Z (in the nonsmooth regions away from contact discontinuities) fifth-order interpolation. Numerical results for the one- and two-dimensional compressible Euler equations show that the proposed adaptive method improves both the computational efficiency and resolution of complex flow features compared with the non-adaptive fifth-order A-WENO scheme.

Figures (11)

• Figure 5.1: Example 1: Density $\rho$ computed by the adaptive and A-WENO scheme with $\Delta x=1/30$ (left) and zoom at $x\in[8.9,14]$ (right).
• Figure 5.2: Example 1: Density $\rho$ computed by the adaptive and A-WENO scheme with $\Delta x=2/87$ and $\Delta x=1/30$, respectively (left) and zoom at $x\in[8.9,14]$ (right).
• Figure 5.3: Example 2: Density $\rho$ computed by the adaptive and A-WENO scheme with $\Delta x=2/105$ and $\Delta x=1/40$, respectively (left) and zoom at $x\in[-0.9,1.6]$ (right).
• Figure 5.4: Example 3: Density $\rho$ (left) and zoom at $x\in[0.56,0.62]$ (right).
• Figure 5.5: Example 4 (Configuration 3): Density $\rho$ computed by the adaptive (top left) and A-WENO (top right) schemes on a uniform mesh with $\Delta x=\Delta y=3/1000$, along with SIs in the $x$-direction (bottom left) and $y$-direction (bottom right).

Theorems & Definitions (6)

• Remark 2.1
• Remark 3.1
• Remark 3.2
• Remark 4.1
• Remark 5.1
• Remark 5.2