A 3D Lagrangian cell-centered hydrodynamic method with higher-order reconstructions for gas and solid dynamics

Abstract The Lagrangian finite volume cell-centered hydrodynamic method introduces dissipation into the calculation by solving a multidirectional approximate Riemann problem. The amount of dissipation created is a function of the jump in the velocity and stress at the node. These jumps in velocity and stress can be reduced by using higher-order reconstructions that are constructed by fitting cell-average values in neighboring cells. One challenge is that a large stencil is required to create high-order reconstructions. To address this challenge, a new two-step reconstruction process is proposed to build a quadratic polynomial by only communicating with face-neighboring cells. The two-step reconstruction method is applied to scalar, vector, and tensor fields. Another challenge is that limiters must be used to reduce the higher order reconstructions towards piecewise constant fields near discontinuities to prevent artificial new extrema. We address this second challenge and propose a new hierarchical limiter that uses a discrete Mach number as a smoothness indicator. The inclusion of the Mach number is essential for minimizing dissipation errors. The new limiter is used with the reconstructions of pressure, velocity, and deviatoric stress. The accuracy and robustness of the new two-step reconstruction method with the new limiter is demonstrated by simulating a suite of 3D Cartesian test problems covering both gas and solid dynamics.