A high-order finite difference method for moving immersed domain boundaries and material interfaces

Gabbard, James; van Rees, Wim M.

doi:10.1016/j.jcp.2024.112979

Title: A high-order finite difference method for moving immersed domain boundaries and material interfaces

Journal Article · 01 April 2024 · Journal of Computational Physics

DOI: https://doi.org/10.1016/j.jcp.2024.112979 · OSTI ID:2338053

Gabbard, James ^[1];

^[2]

Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States); MIT
Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)

Here, we present a high-order sharp treatment of immersed moving domain boundaries and material interfaces, and apply it to the advection-diffusion equation in two and three dimensions. The spatial discretization combines dimension-split finite difference schemes with an immersed boundary treatment based on a weighted least-squares reconstruction of the solution, providing stable discretizations with up to sixth order accuracy for diffusion terms and third order accuracy for advection terms. The temporal discretization relies on a novel strategy for maintaining high-order temporal accuracy in problems with moving boundaries that minimizes implementation complexity and allows arbitrary explicit or diagonally-implicit Runge-Kutta schemes. The approach is broadly compatible with popular PDE-specialized Runge-Kutta time integrators, including low-storage, strong stability preserving, and diagonally implicit schemes. Through numerical experiments we demonstrate that the full discretization maintains high-order spatial and temporal accuracy in the presence of complex 3D geometries and for a range of boundary conditions, including Dirichlet, Neumann, and flux conditions with large jumps in coefficients.

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Research Organization:: Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States); Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES). Scientific User Facilities (SUF)

Grant/Contract Number:: SC0020998; AC02-05CH11231

OSTI ID:: 2338053

Alternate ID(s):: OSTI ID: 2335806

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 507; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (52)

Stability analysis of the inverse Lax–Wendroff boundary treatment for high order upwind-biased finite difference schemes Li, Tingting; Shu, Chi-Wang; Zhang, Mengping Journal of Computational and Applied Mathematics, Vol. 299 https://doi.org/10.1016/j.cam.2015.11.038	journal	June 2016
The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources LeVeque, Randall J.; Li, Zhilin SIAM Journal on Numerical Analysis, Vol. 31, Issue 4, p. 1019-1044 https://doi.org/10.1137/0731054	journal	August 1994
Global optimization of explicit strong-stability-preserving Runge-Kutta methods Ruuth, Steven J. Mathematics of Computation, Vol. 75, Issue 253 https://doi.org/10.1090/S0025-5718-05-01772-2	journal	September 2005
An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries Yang, Jianming; Balaras, Elias Journal of Computational Physics, Vol. 215, Issue 1 https://doi.org/10.1016/j.jcp.2005.10.035	journal	June 2006
Accurate Multilayered Shell Buckling Analysis via the Implicit-Mesh Discontinuous Galerkin Method Guarino, Giuliano; Gulizzi, Vincenzo; Milazzo, Alberto AIAA Journal, Vol. 60, Issue 12 https://doi.org/10.2514/1.J061933	journal	December 2022
A high-order immersed boundary method for acoustic wave scattering and low-Mach number flow-induced sound in complex geometries Seo, Jung Hee; Mittal, Rajat Journal of Computational Physics, Vol. 230, Issue 4 https://doi.org/10.1016/j.jcp.2010.10.017	journal	February 2011
An immersed interface method for simulating the interaction of a fluid with moving boundaries Xu, Sheng; Wang, Z. Jane Journal of Computational Physics, Vol. 216, Issue 2 https://doi.org/10.1016/j.jcp.2005.12.016	journal	August 2006
High-order dimensionally-split Cartesian embedded boundary method for non-dissipative schemes Sharan, Nek; Brady, Peter T.; Livescu, Daniel Journal of Computational Physics, Vol. 464 https://doi.org/10.1016/j.jcp.2022.111341	journal	September 2022
High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles Saye, R. I. SIAM Journal on Scientific Computing, Vol. 37, Issue 2 https://doi.org/10.1137/140966290	journal	January 2015
Design of DIRK schemes with high weak stage order Biswas, Abhijit; Ketcheson, David I.; Seibold, Benjamin Communications in Applied Mathematics and Computational Science, Vol. 18, Issue 1 https://doi.org/10.2140/camcos.2023.18.1	journal	May 2023
Representations of Runge--Kutta Methods and Strong Stability Preserving Methods Higueras, Inmaculada SIAM Journal on Numerical Analysis, Vol. 43, Issue 3 https://doi.org/10.1137/S0036142903427068	journal	January 2005
Development of immersed boundary computational aeroacoustic prediction capabilities for open-rotor noise Brehm, C.; Barad, M. F.; Kiris, C. C. Journal of Computational Physics, Vol. 388 https://doi.org/10.1016/j.jcp.2019.02.011	journal	July 2019
Unstructured Cartesian refinement with sharp interface immersed boundary method for 3D unsteady incompressible flows Angelidis, Dionysios; Chawdhary, Saurabh; Sotiropoulos, Fotis Journal of Computational Physics, Vol. 325 https://doi.org/10.1016/j.jcp.2016.08.028	journal	November 2016
A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries Ge, Liang; Sotiropoulos, Fotis Journal of Computational Physics, Vol. 225, Issue 2 https://doi.org/10.1016/j.jcp.2007.02.017	journal	August 2007
Low-storage implicit/explicit Runge–Kutta schemes for the simulation of stiff high-dimensional ODE systems Cavaglieri, Daniele; Bewley, Thomas Journal of Computational Physics, Vol. 286 https://doi.org/10.1016/j.jcp.2015.01.031	journal	April 2015
A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem Gibou, Frédéric; Fedkiw, Ronald Journal of Computational Physics, Vol. 202, Issue 2 https://doi.org/10.1016/j.jcp.2004.07.018	journal	January 2005
A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries Mittal, R.; Dong, H.; Bozkurttas, M. Journal of Computational Physics, Vol. 227, Issue 10 https://doi.org/10.1016/j.jcp.2008.01.028	journal	May 2008
The Theoretical Accuracy of Runge–Kutta Time Discretizations for the Initial Boundary Value Problem: A Study of the Boundary Error Carpenter, Mark H.; Gottlieb, David; Abarbanel, Saul SIAM Journal on Scientific Computing, Vol. 16, Issue 6 https://doi.org/10.1137/0916072	journal	November 1995
High-Order Immersed-Boundary Method for Incompressible Flows Zhu, Chi; Luo, Haoxiang; Li, Guibo AIAA Journal, Vol. 54, Issue 9 https://doi.org/10.2514/1.J054628	journal	September 2016
A sharp immersed method for 2D flow-body interactions using the vorticity-velocity Navier-Stokes equations Ji, Xinjie; Gabbard, James; van Rees, Wim M. Journal of Computational Physics, Vol. 494 https://doi.org/10.1016/j.jcp.2023.112513	journal	December 2023
A fourth-order Cartesian grid embedded boundary method for Poisson’s equation Devendran, Dharshi; Graves, Daniel; Johansen, Hans Communications in Applied Mathematics and Computational Science, Vol. 12, Issue 1 https://doi.org/10.2140/camcos.2017.12.51	journal	January 2017
Efficient implementation of high order inverse Lax–Wendroff boundary treatment for conservation laws Tan, Sirui; Wang, Cheng; Shu, Chi-Wang Journal of Computational Physics, Vol. 231, Issue 6 https://doi.org/10.1016/j.jcp.2011.11.037	journal	March 2012
An extension and analysis of the Shu-Osher representation of Runge-Kutta methods Ferracina, L.; Spijker, M. N. Mathematics of Computation, Vol. 74, Issue 249 https://doi.org/10.1090/S0025-5718-04-01664-3	journal	June 2004
A systematic approach for constructing higher-order immersed boundary and ghost fluid methods for fluid–structure interaction problems Zeng, Xianyi; Farhat, Charbel Journal of Computational Physics, Vol. 231, Issue 7 https://doi.org/10.1016/j.jcp.2011.12.027	journal	April 2012
A coupled discontinuous Galerkin-Finite Volume framework for solving gas dynamics over embedded geometries Gulizzi, Vincenzo; Almgren, Ann S.; Bell, John B. Journal of Computational Physics, Vol. 450 https://doi.org/10.1016/j.jcp.2021.110861	journal	February 2022
Foundations for high-order, conservative cut-cell methods: Stable discretizations on degenerate meshes Brady, P. T.; Livescu, D. Journal of Computational Physics, Vol. 426 https://doi.org/10.1016/j.jcp.2020.109794	journal	February 2021
An immersed boundary fluid–structure interaction method for thin, highly compliant shell structures Boustani, Jonathan; Barad, Michael F.; Kiris, Cetin C. Journal of Computational Physics, Vol. 438 https://doi.org/10.1016/j.jcp.2021.110369	journal	August 2021
Explicit Runge–Kutta Methods with Estimates of the Local Truncation Error Verner, J. H. SIAM Journal on Numerical Analysis, Vol. 15, Issue 4 https://doi.org/10.1137/0715051	journal	August 1978
A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) Fedkiw, Ronald P.; Aslam, Tariq; Merriman, Barry Journal of Computational Physics, Vol. 152, Issue 2 https://doi.org/10.1006/jcph.1999.6236	journal	July 1999
MURPHY---A Scalable Multiresolution Framework for Scientific Computing on 3D Block-Structured Collocated Grids Gillis, Thomas; van Rees, Wim M. SIAM Journal on Scientific Computing, Vol. 44, Issue 5 https://doi.org/10.1137/21M141676X	journal	September 2022
An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries Le, D. V.; Khoo, B. C.; Peraire, J. Journal of Computational Physics, Vol. 220, Issue 1 https://doi.org/10.1016/j.jcp.2006.05.004	journal	December 2006
An inverse Lax-Wendroff procedure for hyperbolic conservation laws with changing wind direction on the boundary Lu, Jianfang; Shu, Chi-Wang; Tan, Sirui Journal of Computational Physics, Vol. 426 https://doi.org/10.1016/j.jcp.2020.109940	journal	February 2021
A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains Linnick, Mark N.; Fasel, Hermann F. Journal of Computational Physics, Vol. 204, Issue 1 https://doi.org/10.1016/j.jcp.2004.09.017	journal	March 2005
An efficient, high-order method for solving Poisson equation for immersed boundaries: Combination of compact difference and multiscale multigrid methods Hosseinverdi, Shirzad; Fasel, Hermann F. Journal of Computational Physics, Vol. 374 https://doi.org/10.1016/j.jcp.2018.08.006	journal	December 2018
The Correct Formulation of Intermediate Boundary Conditions for Runge--Kutta Time Integration of Initial Boundary Value Problems Pathria, D. SIAM Journal on Scientific Computing, Vol. 18, Issue 5 https://doi.org/10.1137/S1064827594273948	journal	September 1997
High-order quadrature on multi-component domains implicitly defined by multivariate polynomials Saye, Robert I. Journal of Computational Physics, Vol. 448 https://doi.org/10.1016/j.jcp.2021.110720	journal	January 2022
A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies Gilmanov, Anvar; Sotiropoulos, Fotis Journal of Computational Physics, Vol. 207, Issue 2 https://doi.org/10.1016/j.jcp.2005.01.020	journal	August 2005
Flow patterns around heart valves: A numerical method Peskin, Charles S. Journal of Computational Physics, Vol. 10, Issue 2 https://doi.org/10.1016/0021-9991(72)90065-4	journal	October 1972
An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies Gabbard, James; Gillis, Thomas; Chatelain, Philippe Journal of Computational Physics, Vol. 464 https://doi.org/10.1016/j.jcp.2022.111339	journal	September 2022
Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies Borazjani, Iman; Ge, Liang; Sotiropoulos, Fotis Journal of Computational Physics, Vol. 227, Issue 16 https://doi.org/10.1016/j.jcp.2008.04.028	journal	August 2008
A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains Johansen, Hans; Colella, Phillip Journal of Computational Physics, Vol. 147, Issue 1 https://doi.org/10.1006/jcph.1998.5965	journal	November 1998
A Fourth-Order Embedded Boundary Finite Volume Method for the Unsteady Stokes Equations with Complex Geometries Overton-Katz, Nathaniel; Gao, Xinfeng; Guzik, Stephen SIAM Journal on Scientific Computing, Vol. 45, Issue 5 https://doi.org/10.1137/22M1532019	journal	September 2023
Runge–Kutta methods with minimum storage implementations Ketcheson, David I. Journal of Computational Physics, Vol. 229, Issue 5 https://doi.org/10.1016/j.jcp.2009.11.006	journal	March 2010
Low-storage Runge-Kutta schemes Williamson, J. H. Journal of Computational Physics, Vol. 35, Issue 1 https://doi.org/10.1016/0021-9991(80)90033-9	journal	March 1980
A high order moving boundary treatment for compressible inviscid flows Tan, Sirui; Shu, Chi-Wang Journal of Computational Physics, Vol. 230, Issue 15 https://doi.org/10.1016/j.jcp.2011.04.011	journal	July 2011
Inverse Lax–Wendroff procedure for numerical boundary conditions of convection–diffusion equations Lu, Jianfang; Fang, Jinwei; Tan, Sirui Journal of Computational Physics, Vol. 317 https://doi.org/10.1016/j.jcp.2016.04.059	journal	July 2016
Immersed interface methods for moving interface problems Li, Zhilin Numerical Algorithms, Vol. 14, Issue 4, p. 269-293 https://doi.org/10.1023/A:1019173215885	journal	January 1997
An efficient explicit jump high-order compact immersed interface approach for transient incompressible viscous flows Singhal, Raghav; Kalita, Jiten C. Physics of Fluids, Vol. 34, Issue 10 https://doi.org/10.1063/5.0107308	journal	October 2022
A Sharp Interface Cartesian Grid Method for Simulating Flows with Complex Moving Boundaries Udaykumar, H. S.; Mittal, R.; Rampunggoon, P. Journal of Computational Physics, Vol. 174, Issue 1 https://doi.org/10.1006/jcph.2001.6916	journal	November 2001
Computation of Solid–Liquid Phase Fronts in the Sharp Interface Limit on Fixed Grids Udaykumar, H. S.; Mittal, R.; Shyy, Wei Journal of Computational Physics, Vol. 153, Issue 2 https://doi.org/10.1006/jcph.1999.6294	journal	August 1999
Efficient implementation of essentially non-oscillatory shock-capturing schemes Shu, Chi-Wang; Osher, Stanley Journal of Computational Physics, Vol. 77, Issue 2 https://doi.org/10.1016/0021-9991(88)90177-5	journal	August 1988
Modeling wave propagation in elastic solids via high-order accurate implicit-mesh discontinuous Galerkin methods Gulizzi, Vincenzo; Saye, Robert Computer Methods in Applied Mechanics and Engineering, Vol. 395 https://doi.org/10.1016/j.cma.2022.114971	journal	May 2022

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Title: A high-order finite difference method for moving immersed domain boundaries and material interfaces

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