Locally corrected semi-Lagrangian methods for Stokes flow with moving elastic interfaces

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We present a new method for computing two-dimensional Stokes flow with moving interfaces that respond elastically to stretching. The interface is moved by semi-Lagrangian contouring: a distance function is introduced on a tree of cells near the interface, transported by a semi-Lagrangian time step and then used to contour the new interface. The velocity field in a periodic box is calculated as a potential integral resulting from interfacial and body forces, using a technique based on Ewald summation with analytically derived local corrections. The interfacial stretching is found from a surprisingly natural formula. A test problem with an exact solution is constructed and used to verify the speed, accuracy and robustness of the approach. © 2007 Elsevier Inc. All rights reserved.






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Beale, JT, and J Strain (2008). Locally corrected semi-Lagrangian methods for Stokes flow with moving elastic interfaces. Journal of Computational Physics, 227(8). pp. 3896–3920. 10.1016/j.jcp.2007.11.047 Retrieved from https://hdl.handle.net/10161/6958.

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J. Thomas Beale

Professor Emeritus of Mathematics

Here are five recent papers:
J. T. Beale, Solving partial differential equations on closed surfaces with planar Cartesian grids, SIAM J. Sci. Comput. 42 (2020), A1052-A1070 or arxiv.org/abs/1908.01796
S. Tlupova and J. T. Beale, Regularized single and double layer integrals in 3D Stokes flow,  J. Comput. Phys.  386 (2019), 568-584 or arxiv.org/abs/1808.02177
J. T. Beale and W. Ying, Solution of the Dirichlet problem by a finite difference analog of the boundary integral equation, Numer. Math. 141(2019), 605-626 or arxiv.org/abs/1803.08532
J. T. Beale, W. Ying, and J. R. Wilson, A simple method for computing singular or nearly singular integrals on closed surfaces,  Comm. Comput. Phys. 20 (2016), 733-753 or  arxiv.org/abs/1508.00265
J. T. Beale, Uniform error estimates for Navier-Stokes flow with an exact moving boundary using the immersed interface method, SIAM J. Numer. Anal. 53 (2015), 2097-2111 or  arxiv.org/abs/1503.05810

Much of my work has to do with incompressible fluid flow, especially qualitative properties of solutions and behavior of numerical methods, using analytical tools of partial differential equations. My research of the last few years has the dual goals of designing numerical methods for problems with interfaces, especially moving interfaces in fluid flow, and the analysis of errors in computational methods of this type. We have developed a general method for the numerical computation of singular or nearly singular integrals, such as layer potentials on a curve or surface, evaluated at a point on the curve or surface or nearby, in work with M.-C. Lai, A. Layton, S. Tlupova, and W. Ying. After regularizing the integrand, a standard quadrature is used, and corrections are added which are determined analytically. Current work with coworkers is intended to make these methods more practical, especially in three dimensional simulations. Some projects (partly with Anita Layton) concern the design of numerical methods which combine finite difference methods with separate computations on interfaces. We developed a relatively simple approach for computing Navier-Stokes flow with an elastic interface. In analytical work we have derived estimates in maximum norm for elliptic (steady-state) and parabolic (diffusive) partial differential equations. For problems with interfaces, maximum norm estimates are more informative than the usual ones in the L^2 sense. More general estimates were proved by Michael Pruitt in his Ph.D. thesis.

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