Abstract
Discontinuous Galerkin (DG) method is a popular high-order accurate method for solving unsteady convection-dominated problems. After spatially discretizing the problem with the DG method, a time integration scheme is necessary for evolving the result. Owing to the stability-based restriction, the time step for an explicit scheme is limited by the smallest element size within the mesh, making the calculation inefficient. In this paper, a hybrid scheme comprising a three-stage, third-order accurate, and strong stability preserving Runge–Kutta (SSP-RK3) scheme and the three-stage, third-order accurate, L-stable, and diagonally implicit Runge–Kutta (LDIRK3) scheme is proposed. By dealing with the coarse and the refined elements with the explicit and implicit schemes, respectively, the time step for the hybrid scheme is free from the limitation of the smallest element size, making the simulation much more efficient. Numerical tests and comparison studies were made to show the performance of the hybrid scheme.
Highlights
The convection-dominated problem plays a key role in fluid dynamics, mass and heat transfer, and many other fields
We found that the convergence order in space is N + 1, which is optional for the discontinuous Galerkin (DG) method in solving conservation laws [6]
Similar to the 1D cases, the optimal (N + 1)th order for the DG method is achieved. It indicates that the high-order accuracy of the SSP-RK3 scheme or the LDIRK3 scheme is maintained by the hybrid scheme, and the impact of the big size changes in the mesh is negligible
Summary
The convection-dominated problem plays a key role in fluid dynamics, mass and heat transfer, and many other fields. The DG method was used to spatially discretize the governing equation; a time integration scheme was used to evolve the result in the time domain. This was followed by many other studies. In [11], after spatially discretizing the Maxwell’s equations with the DG method, a combination of the explicit leap-frog scheme and the implicit Crank–Nicolson scheme was used to integrate the result. We develop a new hybrid time integration scheme comprising two high-order schemes for the evolution of the DG discretization.
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