Journal of Computational Physics
Journal of Computational Physics
A Moving Mesh Method Based on the Geometric Conservation Law
SIAM Journal on Scientific Computing
Journal of Computational Physics
Solving unsymmetric sparse systems of linear equations with PARDISO
Future Generation Computer Systems - Special issue: Selected numerical algorithms
Journal of Computational Physics
Journal of Computational Physics
Discontinuous Galerkin spectral element approximations on moving meshes
Journal of Computational Physics
Boundary states at reflective moving boundaries
Journal of Computational Physics
Some manufactured solutions for verification of fluid-structure interaction codes
Computers and Structures
High-Order Local Time Stepping on Moving DG Spectral Element Meshes
Journal of Scientific Computing
ALE-DGSEM approximation of wave reflection and transmission from a moving medium
Journal of Computational Physics
Hi-index | 31.47 |
This paper takes a fresh look at the geometric conservation law (GCL) from the perspective of the finite element method (FEM) for incompressible flows. The GCL arises naturally in the context of Arbitrary Lagrangian Eulerian (ALE) formulations for solving problems on deforming domains. GCL compliance is traditionally interpreted as a consistency criterion for applying an unsteady flow solution algorithm to simulate exactly a uniform flow on a deforming domain. We introduce an additional requirement: the time integrator must maintain its fixed mesh accuracy when applied to deforming meshes. A review of the literature shows that while many authors use an ALE FEM, few of them discuss the GCL issues. We show how a fixed mesh unsteady FEM using high order time integrator (up to fifth order in time) can be transposed to solve problems on deforming meshes and preserve its fixed mesh high order temporal accuracy. An appropriate construction of the divergence of the mesh velocity guarantees GCL compliance while a separate construction of the mesh velocity itself allows the time-integrator to deliver its fixed mesh high order temporal accuracy on deforming domains. Analytical error analysis of problems with closed form solutions provides insight on the behavior of the time integrators. It also explains why high order temporal accuracy is achieved with a conservative formulation of the incompressible Navier-Stokes equations, while only first order time accuracy is observed with the non-conservative formulation and all time-integrators investigated here. We present thorough time-step and grid refinement studies for simple problems with closed form solutions and for a manufactured solution with a non-trivial flow on a deforming mesh. In all cases studied, the proposed reconstructions of the mesh velocity and its divergence for the conservative formulation lead to optimal time accuracy on deforming grids.