A high-order SRCR-DG method for simulating viscoelastic flows at high Weissenberg numbers

The high Weissenberg number problem (HWNP) has been regarded as a key challenge in computational rheology. Developing stable and flexible numerical methods for simulating highly elastic fluid flows has become the major research subject of scholars. In this work, we present a new stabilized high-order discontinuous Galerkin (DG) method to simulate viscoelastic flows at high Weissenberg (Wi) numbers. To decouple the velocity and pressure, and circumvent the limitation of the Ladyzhenskaya–Babuška–Brezzi condition, the dual splitting scheme based on the implicit backward differentiation formula and explicit extrapolation scheme is performed for the momentum and mass equations. In addition, integration by parts of the pressure gradient term in projection step and the velocity divergence term in pressure Poisson equation is required to enhance the numerical stability of the method at small time steps. To guarantee the positive definiteness of the conformation tensor and improve the computational stability at high Wi numbers, the square-root-conformation representation (SRCR) approach is applied to reconstruct the constitutive equation. Meanwhile, the second-order Runge–Kutta scheme is performed for temporal discretization and the local Lax–Friedrichs flux is employed for the nonlinear convective term to tackle the convection-dominated problems. The resulting numerical method is free from iterative errors, and does not require any additional stabilization term, which gives an efficient and easy-to-be-implemented solver for viscoelastic flows. The numerical study considers different viscoelastic flow problems, including the Poiseuille flow in a channel, the lid-driven cavity flow, the flow past a fixed cylinder and the flow around an oscillating cylinder. The numerical results reveal the high-order accuracy and robustness of our method at high Wi numbers. Compared with the scheme in the literature, the presented method is more flexible in complex irregular regions and much easier to construct the high-order format. This is the first attempt to apply SRCR scheme in framework of DG method, which provides a new efficient high-order method for simulating complex viscoelastic flows at high Wi numbers.