Alternating Polynomial Reconstruction Method for Hyperbolic Conservation Laws

We propose a new multi-moment numerical solver for hyperbolic conservation laws by using the alternating polynomial reconstruction approach. Unlike existing multi-moment schemes, our approach updates model variables by implementing two polynomial reconstructions alternately. First, Hermite interpolation reconstructs the solution within the cell by matching the point-based variables containing both physical values and their spatial derivatives. Then the reconstructed solution is updated by the Euler method. Second, we solve a constrained least-squares problem to correct the updated solution to preserve the conservation laws. Our method enjoys the advantages of a compact numerical stencil and high-order accuracy. Fourier analysis also indicates that our method allows a larger CFL number compared with many other high-order schemes. By adding a proper amount of artificial viscosity, shock waves and other discontinuities can also be computed accurately and sharply without solving an approximated Riemann problem.

Automated summary

The new multi-moment numerical solver proposed in this study utilizes alternating polynomial reconstruction to solve hyperbolic conservation laws. This method allows for a compact numerical stencil and high-order accuracy, with the ability to accommodate a larger CFL number compared to other high-order schemes. By incorporating artificial viscosity, shock waves and discontinuities can be accurately computed without the need to solve an approximated Riemann problem.