Volume 13, Issue 3
A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions

Yaning Xie & Wenjun Ying

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 595-619.

Published online: 2020-03

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  • Abstract

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.

  • Keywords

Unsteady Stokes equations, Navier-Stokes equations, stream function-vorticity formulation, kernel-free boundary integral method, composite backward difference formula, semi-implicit Runge-Kutta method.

  • AMS Subject Headings

52B10, 65D18, 68U05, 68U07

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

wying@sjtu.edu.cn (Wenjun Ying)

  • BibTex
  • RIS
  • TXT
@Article{NMTMA-13-595, author = {Xie , Yaning and Ying , Wenjun}, title = {A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {13}, number = {3}, pages = {595--619}, abstract = {

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0175}, url = {http://global-sci.org/intro/article_detail/nmtma/15777.html} }
TY - JOUR T1 - A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions AU - Xie , Yaning AU - Ying , Wenjun JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 595 EP - 619 PY - 2020 DA - 2020/03 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0175 UR - https://global-sci.org/intro/article_detail/nmtma/15777.html KW - Unsteady Stokes equations, Navier-Stokes equations, stream function-vorticity formulation, kernel-free boundary integral method, composite backward difference formula, semi-implicit Runge-Kutta method. AB -

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.

Yaning Xie & Wenjun Ying. (2020). A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions. Numerical Mathematics: Theory, Methods and Applications. 13 (3). 595-619. doi:10.4208/nmtma.OA-2019-0175
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