High Order Multi-dimensional Characteristics Tracing for the Incompressible Euler Equation and the Guiding-Center Vlasov Equation

In this paper, we propose a high order characteristics tracing scheme for the two-dimensional nonlinear incompressible Euler system in vorticity stream function formulation and the guiding center Vlasov model. Such a scheme is incorporated into a semi-Lagrangian finite difference WENO framework for simulating the aforementioned model equations. This is an extension of our earlier work on high order characteristics tracing scheme for the 1D nonlinear Vlasov–Poisson system (Qiu and Russo in J Sci Comput 71:414–434, 2017). The effectiveness of the proposed scheme is demonstrated numerically by an extensive set of test cases.

[1]  Louis F. Rossi,et al.  Merging Computational Elements in Vortex Simulations , 1997, SIAM J. Sci. Comput..

[2]  Eric Sonnendrücker,et al.  Conservative semi-Lagrangian schemes for Vlasov equations , 2010, J. Comput. Phys..

[3]  Roberto Ferretti,et al.  A fully semi-Lagrangian discretization for the 2D incompressible Navier-Stokes equations in the vorticity-streamfunction formulation , 2018, Appl. Math. Comput..

[4]  Alain Ghizzo,et al.  An Eulerian code for the study of the drift-kinetic Vlasov equation , 1993 .

[5]  Centro internazionale matematico estivo. Session,et al.  Advanced Numerical Approximation of Nonlinear Hyperbolic Equations , 1998 .

[6]  A. Chorin.  Numerical study of slightly viscous flow , 1973, Journal of Fluid Mechanics.

[7]  Jian‐Guo Liu,et al.  High order finite difference methods for unsteady incompressible flows in multi-connected domains , 2004 .

[8]  Chi-Wang Shu,et al.  A High-Order Discontinuous Galerkin Method for 2D Incompressible Flows , 2000 .

[9]  Chi-Wang Shu,et al.  Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow , 2011, J. Comput. Phys..

[10]  Wei Guo,et al.  A high order semi-Lagrangian discontinuous Galerkin method for Vlasov-Poisson simulations without operator splitting , 2017, J. Comput. Phys..

[11]  Tao Xiong,et al.  Conservative Multi-dimensional Semi-Lagrangian Finite Difference Scheme: Stability and Applications to the Kinetic and Fluid Simulations , 2019, J. Sci. Comput..

[12]  Maxim A. Olshanskii,et al.  Natural vorticity boundary conditions on solid walls , 2015 .

[13]  Elena Celledoni,et al.  High Order Semi-Lagrangian Methods for the Incompressible Navier–Stokes Equations , 2012, J. Sci. Comput..

[14]  E. Sonnendrücker,et al.  The Semi-Lagrangian Method for the Numerical Resolution of the Vlasov Equation , 1999 .

[15]  Giovanni Russo,et al.  A deterministic vortex method for the Navier-Stokes equations , 1993 .

[16]  Petros Koumoutsakos,et al.  Vortex Methods: Theory and Practice , 2000 .

[17]  A. Leonard.  Vortex methods for flow simulation , 1980 .

[18]  M. Falcone,et al.  Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations , 2014 .

[19]  E Weinan,et al.  Essentially Compact Schemes for Unsteady Viscous Incompressible Flows , 1996 .

[20]  Giovanni Russo,et al.  Fast triangulated vortex methods for the 2D Euler equations , 1994 .

[21]  Jianxian Qiu,et al.  An h-Adaptive RKDG Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model , 2017, J. Sci. Comput..

[22]  R. Krasny.  Desingularization of periodic vortex sheet roll-up , 1986 .

[23]  Jian‐Guo Liu,et al.  Vorticity Boundary Condition and Related Issues for Finite Difference Schemes , 1996 .

[24]  George Em Karniadakis,et al.  A semi-Lagrangian high-order method for Navier-Stokes equations , 2001 .

[25]  Magdi Shoucri.  A two-level implicit scheme for the numerical solution of the linearized vorticity equation , 1981 .

[26]  Vorticity boundary conditions for Navier-Stokes equations , 1996 .

[27]  Giovanni Russo,et al.  A High Order Multi-Dimensional Characteristic Tracing Strategy for the Vlasov–Poisson System , 2017, J. Sci. Comput..