High-Order Linearly Implicit Two-Step Peer Methods for the Discontinuous Galerkin Solution of the Incompressible Rans Equations

Abstract

Abstract. In recent years the increasing attention to high-order Finite Volume (FV), Finite Element (FE) and spectral methods and the growth of computing power promote the development of high-order temporal schemes to perform robust, accurate and efficient unsteady long-time simulations. In this context, some features of the Discontinuous Galerkin finite element (DG) methods, e.g. compactness and flexibility, can be advantageous both for explicit and implicit time integration approaches. Explicit schemes can achieve very high accuracy, but are limited by time-step restrictions, while implicit schemes, even if memory consuming, can overcome time-step limitations, thus improving the time integration efficiency. During last decades several high order implicit temporal schemes have been developed, and some of them have been successfully coupled with DG methods. However these schemes can show the order reduction if applied to very stiff problems or problems with time-dependent boundary conditions. To overcome these limitations, high-order linearly implicit two-step peer methods have been proposed and successfully applied to the numerical solution of differential-algebraic equations. The aim of the present work is to implement high-order two-step peer methods in a DG code and assess their performance for the unsteady solution of the incompressible Navier-Stokes (INS) and Reynolds Averaged Navier-Stokes equations (RANS) equations.

Assessment of an adaptive time integration strategy for a high‐order discretization of the unsteady RANS equations

“…Several test cases with increasing stiffness and difficulty are considered: the turbulent flow around a circular cylinder at two different Reynolds numbers $$$ \mathit{\operatorname{Re}}=\left\{3900,140,000\right\} $$$, the flow around a tandem cylinders at $$$ \mathit{\operatorname{Re}}=166,000 $$$, and, finally, the turbulent flow through a VAWT at $$$ \mathit{\operatorname{Re}}=160,000 $$$ 16,17 . The time integration schemes used are: the linearly implicit one‐step Rosenbrock‐type Runge–Kutta of third order/three stages ROS3PL, 9 fourth order/six stages RODASP, 10 and fifth order/eight stages ROD5_1, 11 and the linearly implicit Rosenbrock‐type two‐step peer schemes 6,12 . The controllers applied within the adaptation strategy are: the standard, 13 the PI 4.2, 14 and the H211b and H312b 15 .…”

“…16,17 The time integration schemes used are: the linearly implicit one-step Rosenbrock-type Runge-Kutta of third order/three stages ROS3PL, 9 fourth order/six stages RODASP, 10 and fifth order/eight stages ROD5_1, 11 and the linearly implicit Rosenbrock-type two-step peer schemes. 6,12 The controllers applied within the adaptation strategy are: the standard, 13 the PI 4.2, 14 and the H211b and H312b. 15 In addition, a smooth limiter function, a correction factor and a maximum step size limit are introduced.…”