Adaptive time stepping for vesicle suspensions

We present an adaptive high-order accurate time-stepping numerical scheme for the flow of vesicles suspended in Stokesian fluids. Our scheme can be summarized as an approximate implicit spectral deferred correction (SDC) method. Applying a fully-implicit SDC scheme to vesicle flows is prohibitively expensive. For this reason we introduce several approximations. Our scheme is based on a semi-implicit linearized low-order time stepping method. (Our discretization is spectrally accurate in space.) We expect that the accuracy can be arbitrary-order, but our examples suffer from order reduction which limits the observed accuracy to third-order. We also use invariant properties of vesicle flows, constant area and boundary length in two dimensions, to reduce the computational cost of error estimation for adaptive time stepping. We present results in two dimensions for single-vesicle flows, constricted geometry flows, converging flows, and flows in a Couette apparatus. We experimentally demonstrate that the proposed scheme enables automatic selection of the time step size and high-order accuracy.

[1]  Chaouqi Misbah,et al.  Vacillating breathing and tumbling of vesicles under shear flow. , 2006, Physical review letters.

[2]  Ming-Chih Lai,et al.  An Unconditionally Energy Stable Penalty Immersed Boundary Method for Simulating the Dynamics of an Inextensible Interface Interacting with a Solid Particle , 2015, J. Sci. Comput..

[3]  T. Biben,et al.  Steady to unsteady dynamics of a vesicle in a flow. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Axel Voigt,et al.  Dynamics of multicomponent vesicles in a viscous fluid , 2010, J. Comput. Phys..

[5]  Jingfang Huang,et al.  Accelerating the convergence of spectral deferred correction methods , 2006, J. Comput. Phys..

[6]  C. Pozrikidis,et al.  The axisymmetric deformation of a red blood cell in uniaxial straining Stokes flow , 1990, Journal of Fluid Mechanics.

[7]  George Biros,et al.  Boundary integral method for the flow of vesicles with viscosity contrast in three dimensions , 2015, J. Comput. Phys..

[8]  Rolf Krause,et al.  A space-time parallel solver for the three-dimensional heat equation , 2013, PARCO.

[9]  Jian Zhang,et al.  Adaptive Finite Element Method for a Phase Field Bending Elasticity Model of Vesicle Membrane Deformations , 2008, SIAM J. Sci. Comput..

[10]  George Biros,et al.  Author ' s personal copy Dynamic simulation of locally inextensible vesicles suspended in an arbitrary two-dimensional domain , a boundary integral method , 2010 .

[11]  Klaus Kassner,et al.  Phase-field approach to three-dimensional vesicle dynamics. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Pierre Saramito,et al.  Computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods , 2014, J. Comput. Phys..

[13]  Ming-Chih Lai,et al.  Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method , 2010, J. Comput. Phys..

[14]  Hong Zhao,et al.  The dynamics of a vesicle in simple shear flow , 2011, Journal of Fluid Mechanics.

[15]  Elizabeth L. Bouzarth,et al.  A multirate time integrator for regularized Stokeslets , 2010, J. Comput. Phys..

[16]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[17]  Andrew J. Christlieb,et al.  Parallel Semi-Implicit Time Integrators , 2012, ArXiv.

[18]  Martin Weiser,et al.  Faster SDC convergence on non-equidistant grids by DIRK sweeps , 2015 .

[19]  George Biros,et al.  A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D , 2009, J. Comput. Phys..

[20]  Jingfang Huang,et al.  Semi-implicit Krylov deferred correction methods for differential algebraic equations , 2012, Math. Comput..

[21]  Richard W. Vuduc,et al.  Petascale Direct Numerical Simulation of Blood Flow on 200K Cores and Heterogeneous Architectures , 2010, 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis.

[22]  Klaus Böhmer,et al.  Defect Correction Methods , 1984, Computing Supplementum.

[23]  Michael L. Minion,et al.  TOWARD AN EFFICIENT PARALLEL IN TIME METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS , 2012 .

[24]  Colin B. Macdonald,et al.  Revisionist integral deferred correction with adaptive step-size control , 2013, 1310.6331.

[25]  H. Noguchi,et al.  Shape transitions of fluid vesicles and red blood cells in capillary flows. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[26]  Matthias Bolten,et al.  Interweaving PFASST and Parallel Multigrid , 2015, SIAM J. Sci. Comput..

[27]  H. Stetter.  Analysis of Discretization Methods for Ordinary Differential Equations , 1973 .

[28]  Steven J. Ruuth,et al.  Implicit-explicit methods for time-dependent partial differential equations , 1995 .

[29]  Robert Speck,et al.  A high-order Boris integrator , 2014, J. Comput. Phys..

[30]  Hong Zhao,et al.  The dynamics of a non-dilute vesicle suspension in a simple shear flow , 2013, Journal of Fluid Mechanics.

[31]  M. Minion.  Semi-implicit spectral deferred correction methods for ordinary differential equations , 2003 .

[32]  Jingfang Huang,et al.  Arbitrary order Krylov deferred correction methods for differential algebraic equations , 2007, J. Comput. Phys..

[33]  George Biros,et al.  Vesicle migration and spatial organization driven by flow line curvature. , 2011, Physical review letters.

[34]  Axel Voigt,et al.  Diffuse interface models of locally inextensible vesicles in a viscous fluid , 2013, J. Comput. Phys..

[35]  Rolf Krause,et al.  Inexact spectral deferred corrections , 2016 .

[36]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[37]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[38]  Anders C. Hansen,et al.  On the order of deferred correction , 2011 .

[39]  L. Greengard,et al.  Spectral Deferred Correction Methods for Ordinary Differential Equations , 2000 .

[40]  Andrew J. Christlieb,et al.  A Parallel Space-Time Algorithm , 2012, SIAM J. Sci. Comput..

[41]  Colin B. Macdonald,et al.  Parallel High-Order Integrators , 2010, SIAM J. Sci. Comput..

[42]  George Biros,et al.  A fast algorithm for simulating vesicle flows in three dimensions , 2011, J. Comput. Phys..

[43]  K. Bohmer.  Defect Correction Methods: Theory and Applications , 1984 .

[44]  C. Misbah,et al.  Vesicles in haptotaxis with hydrodynamical dissipation , 2003, The European physical journal. E, Soft matter.

[45]  E. Sackmann,et al.  Supported Membranes: Scientific and Practical Applications , 1996, Science.

[46]  Bradley K. Alpert,et al.  Hybrid Gauss-Trapezoidal Quadrature Rules , 1999, SIAM J. Sci. Comput..

[47]  Andrew J. Christlieb,et al.  Implicit Parallel Time Integrators , 2011, J. Sci. Comput..

[48]  Jingfang Huang,et al.  Semi-implicit Krylov deferred correction methods for ordinary differential equations , 2009 .

[49]  Rolf Krause,et al.  A multi-level spectral deferred correction method , 2013, BIT Numerical Mathematics.

[50]  Benjamin W. Ong,et al.  COMMENTS ON HIGH-ORDER INTEGRATORS EMBEDDED WITHIN INTEGRAL DEFERRED CORRECTION METHODS , 2009 .

[51]  Jingfang Huang,et al.  Krylov deferred correction accelerated method of lines transpose for parabolic problems , 2008, J. Comput. Phys..

[52]  Saroja Ramanujan,et al.  Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of fluid viscosities , 1998, Journal of Fluid Mechanics.

[53]  Werner Liniger,et al.  Stability of Two-Step Methods for Variable Integration Steps , 1983 .

[54]  Robert D. Skeel,et al.  A Theoretical Framework for Proving Accuracy Results for Deferred Corrections , 1982 .

[55]  Yang Xiang,et al.  An integral equation method for epitaxial step-flow growth simulations , 2006, J. Comput. Phys..

[56]  U. Seifert,et al.  Influence of shear flow on vesicles near a wall: A numerical study. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[57]  T. Biben,et al.  Tumbling of vesicles under shear flow within an advected-field approach. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[58]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[59]  Bengt Lindberg.  Error estimation and iterative improvement for discretization algorithms , 1980 .

[60]  SEBASTIANO BOSCARINO.  Error Analysis of IMEX Runge-Kutta Methods Derived from Differential-Algebraic Systems , 2007, SIAM J. Numer. Anal..

[61]  George Biros,et al.  High-volume fraction simulations of two-dimensional vesicle suspensions , 2013, J. Comput. Phys..

[62]  Rolf Krause,et al.  Inexact spectral deferred corrections using single-cycle multigrid , 2014 .

[63]  A. Bourlioux,et al.  High-order multi-implicit spectral deferred correction methods for problems of reactive flow , 2003 .