Revisiting the 2nd-order semi-discrete finite volume schemes for hyperbolic conservation laws on Cartesian grids

The semi-discrete schemes for hyperbolic conservation laws on Cartesian grids are revisited. For the spatial discretisation the traditional 2nd-order slope limited MUSCL schemes are used. We focus on the explicit Runge-Kutta (RK) type methods of order two and three for the temporal discretisation, including those possessing the strong stability preserving (SSP) property. The qualitative and quantitative influence on the computed solutions of the effective spatial stencil size stemming from particular time integrators is analysed. The reduction of the spatial stencils size by freezing the MUSCL slopes for the entire time step is shown to significantly improve the stability and accuracy of the 2nd-order RK methods: SSPRK22 and midpoint rule (MIP) method. The MIP method with frozen slopes (FS), which can be interpreted as a semi-discrete, unstaggered variant of the MUSCL-Hancock scheme, can be used beyond the hyperbolic problems and is shown to be attractive alternative to SSPRK methods. Importantly, this holds for a wide range of the Courant numbers, including values close to the stability limit, while at low Courant numbers the performance is very similar to SSPRK-based schemes. On the other hand, the MIP method does not exhibit the SSP property; therefore we propose a new one-parameter family of continuously differentiable 2nd-order total variation diminishing slope limiters expressed by a polynomial, which allows to control the required level of numerical diffusion to reduce (or eliminate) the spurious oscillations. Within the use of the new slope limiter no stability issues were found when MIPFS method was applied to a demanding fluid flow simulation involving strong shocks. When a very low Mach number flow is considered, the MIPFS method requires less diffusive slope limiter than SSPRK33 to provide convergence. Importantly, this improvement is obtained when simple and universal spatial discretisation is applied: incomplete approximate Riemann solver of HLL-type along with low-Mach corrected MUSCL reconstruction of velocity.