A GPU-accelerated adaptive discontinuous Galerkin method for level set equation

ABSTRACT This paper presents a GPU-accelerated nodal discontinuous Galerkin method for the solution of two- and three-dimensional level set (LS) equation on unstructured adaptive meshes. Using adaptive mesh refinement, computations are localised mostly near the interface location to reduce the computational cost. Small global time step size resulting from the local adaptivity is avoided by local time-stepping based on a multi-rate Adams–Bashforth scheme. Platform independence of the solver is achieved with an extensible multi-threading programming API that allows runtime selection of different computing devices (GPU and CPU) and different threading interfaces (CUDA, OpenCL and OpenMP). Overall, a highly scalable, accurate and mass conservative numerical scheme that preserves the simplicity of LS formulation is obtained. Efficiency, performance and local high-order accuracy of the method are demonstrated through distinct numerical test cases.

[1]  R. LeVeque.  High-resolution conservative algorithms for advection in incompressible flow , 1996 .

[2]  R Gandham,et al.  GPU Accelerated Discontinuous Galerkin Methods for Shallow Water Equations , 2014, 1403.1661.

[3]  Axel Modave,et al.  A nodal discontinuous Galerkin method for reverse-time migration on GPU clusters , 2015, 1506.00907.

[4]  Jean-François Remacle,et al.  A quadrature-free discontinuous Galerkin method for the level set equation , 2006, J. Comput. Phys..

[5]  P. Colella,et al.  An Adaptive Level Set Approach for Incompressible Two-Phase Flows , 1997 .

[6]  Mark Sussman,et al.  A Discontinuous Spectral Element Method for the Level Set Equation , 2003, J. Sci. Comput..

[7]  Frank Losasso,et al.  A fast and accurate semi-Lagrangian particle level set method , 2005 .

[8]  Tim Warburton,et al.  An explicit construction of interpolation nodes on the simplex , 2007 .

[9]  David Medina,et al.  OKL: A Unified Language for Parallel Architectures , 2015 .

[10]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[11]  Jean-François Remacle,et al.  Multirate time stepping for accelerating explicit discontinuous Galerkin computations with application to geophysical flows , 2013 .

[12]  Mark Ainsworth,et al.  Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods , 2004 .

[13]  Clint Dawson,et al.  A parallel local timestepping Runge-Kutta discontinuous Galerkin method with applications to coastal ocean modeling , 2013 .

[14]  M. Sussman,et al.  A Coupled Level Set and Volume-of-Fluid Method for Computing 3D and Axisymmetric Incompressible Two-Phase Flows , 2000 .

[15]  George Em Karniadakis,et al.  De-aliasing on non-uniform grids: algorithms and applications , 2003 .

[16]  Danping Peng,et al.  Weighted ENO Schemes for Hamilton-Jacobi Equations , 1999, SIAM J. Sci. Comput..

[17]  C. W. Gear,et al.  Multirate linear multistep methods , 1984 .

[18]  Markus Clemens,et al.  GPU Accelerated Adams–Bashforth Multirate Discontinuous Galerkin FEM Simulation of High-Frequency Electromagnetic Fields , 2010, IEEE Transactions on Magnetics.

[19]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[20]  Clint Dawson,et al.  High Resolution Schemes for Conservation Laws with Locally Varying Time Steps , 2000, SIAM J. Sci. Comput..

[21]  Timothy C. Warburton,et al.  Nodal discontinuous Galerkin methods on graphics processors , 2009, J. Comput. Phys..

[22]  J. Hesthaven,et al.  Nodal high-order methods on unstructured grids , 2002 .

[23]  T. Koornwinder.  Two-Variable Analogues of the Classical Orthogonal Polynomials , 1975 .

[24]  Timothy C. Warburton,et al.  OCCA: A unified approach to multi-threading languages , 2014, ArXiv.

[25]  Timothy C. Warburton,et al.  High-Order Finite-differences on multi-threaded architectures using OCCA , 2014, ArXiv.

[26]  Chi-Wang Shu,et al.  High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes , 2003, SIAM J. Sci. Comput..

[27]  Mark Sussman,et al.  A hybrid level set-volume constraint method for incompressible two-phase flow , 2012, J. Comput. Phys..

[28]  Wang Hai-bing,et al.  High-order essentially non-oscillatory schemes for Hamilton-Jacobi equations , 2006 .

[29]  Moshe Dubiner.  Spectral methods on triangles and other domains , 1991 .

[30]  Stanley Osher,et al.  Simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension , 2006, J. Comput. Phys..

[31]  Martin Fuhry,et al.  Discontinuous Galerkin methods on graphics processing units for nonlinear hyperbolic conservation laws , 2016, ArXiv.

[32]  Emil M. Constantinescu,et al.  Multirate Timestepping Methods for Hyperbolic Conservation Laws , 2007, J. Sci. Comput..

[33]  S. Osher,et al.  High-order essentially nonsocillatory schemes for Hamilton-Jacobi equations , 1990 .

[34]  Emil M. Constantinescu,et al.  Multirate Explicit Adams Methods for Time Integration of Conservation Laws , 2009, J. Sci. Comput..

[35]  Marcus J. Grote,et al.  High-order explicit local time-stepping methods for damped wave equations , 2011, J. Comput. Appl. Math..

[36]  Stanley Osher,et al.  Essentially Non-Oscillatory Adaptive Tree Methods , 2008, J. Sci. Comput..

[37]  Willem Hundsdorfer,et al.  Monotonicity-Preserving Linear Multistep Methods , 2003, SIAM J. Numer. Anal..

[38]  Mark Sussman,et al.  An Efficient, Interface-Preserving Level Set Redistancing Algorithm and Its Application to Interfacial Incompressible Fluid Flow , 1999, SIAM J. Sci. Comput..

[39]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[40]  Ian M. Mitchell,et al.  A hybrid particle level set method for improved interface capturing , 2002 .

[41]  J. Sethian,et al.  FRONTS PROPAGATING WITH CURVATURE DEPENDENT SPEED: ALGORITHMS BASED ON HAMILTON-JACOB1 FORMULATIONS , 2003 .