A method-of-lines framework for energy stable arbitrary Lagrangian-Eulerian methods

by   Tomas Lundquist, et al.
Linköping University
University of Cape Town

We present a novel framework based on semi-bounded spatial operators for analyzing and discretizing initial boundary value problems on moving and deforming domains. This development extends an existing framework for well-posed problems and energy stable discretizations from stationary domains to the general case including arbitrary mesh motion. In particular, we show that an energy estimate derived in the physical coordinate system is equivalent to a semi-bounded property with respect to a stationary reference domain. The continuous analysis leading up to this result is based on a skew-symmetric splitting of the material time derivative, and thus relies on the property of integration-by-parts. Following this, a mimetic energy stable arbitrary Lagrangian-Eulerian framework for semi-discretization is formulated, based on approximating the material time derivative in a way consistent with discrete summation-by-parts. Thanks to the semi-bounded property, a method-of-lines approach using standard explicit or implicit time integration schemes can be applied to march the system forward in time. The same type of stability arguments applies as for the corresponding stationary domain problem, without regards to additional properties such as discrete geometric conservation. As an additional bonus we demonstrate that discrete geometric conservation, in the sense of exact free-stream preservation, can still be achieved in an automatic way with the new framework. However, we stress that this is not necessary for stability.


page 1

page 2

page 3

page 4


Parallel energy stable phase field simulations of Ni-based alloys system

In this paper, we investigate numerical methods for solving Nickel-based...

Energy stable schemes for gradient flows based on the DVD method

The existing discrete variational derivative method is only second-order...

Energy Stability of Explicit Runge-Kutta Methods for Non-autonomous or Nonlinear Problems

Many important initial value problems have the property that energy is n...

Semi-Lagrangian Finite-Element Exterior Calculus for Incompressible Flows

We develop a mesh-based semi-Lagrangian discretization of the time-depen...

A Generalized Eulerian-Lagrangian Discontinuous Galerkin Method for Transport Problems

We propose a generalized Eulerian-Lagrangian (GEL) discontinuous Galerki...

Explicit Jacobian matrix formulas for entropy stable summation-by-parts schemes

Entropy stable schemes replicate an entropy inequality at the semi-discr...

Please sign up or login with your details

Forgot password? Click here to reset