Accuracy Enhancing Interface Treatment Algorithm: The Back and Forth Error Compensation and Correction Method

The accuracy of information transmission while solving domain decomposed problems is crucial to smooth transition of a solution around the interface/overlapping region. This paper describes a systematical study on an accuracy enhancing interface treatment algorithm based on the back and forth error compensation and correction method (BFECC). By repetitively employing a low order interpolation technique (normally local 2 order) 3 times, this algorithm achieves local 3 order accuracy. Analytical derivations for 1D & 2D cases are made, and the “super convergence” phenomenon (4 order accuracy) is found for specific positioning of the donor and target grids. A set of numerical experiments based on various relative displacements, relative rotations, mesh ratios, and meshes with perturbation have been tested, and the results match the derivations. Different interface treatments are compared with 3D examples: corner flow and cavity flow. The component convergence rate analysis shows that the BFECC method positively affects the accuracy of solutions.

[1]  Fotis Sotiropoulos,et al.  An overset-grid method for 3D unsteady incompressible flows , 2003 .

[2]  H. S. Tang,et al.  An overset grid method for integration of fully 3D fluid dynamics and geophysics fluid dynamics models to simulate multiphysics coastal ocean flows , 2014, J. Comput. Phys..

[3]  Han-song Tang.  Study on a grid interface algorithm for solutions of incompressible Navier–Stokes equations , 2006 .

[4]  Chaowei Hu,et al.  No . 98-32 Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes , 1998 .

[5]  Yao Li,et al.  A limiting strategy for the back and forth error compensation and correction method for solving advection equations , 2016, Math. Comput..

[6]  Y. Vassilevski,et al.  A semi-Lagrangian method on dynamically adapted octree meshes , 2015 .

[7]  Qinjun Kang,et al.  Lattice Boltzmann method on quadtree grids. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  W. Henshaw,et al.  Composite overlapping meshes for the solution of partial differential equations , 1990 .

[9]  Ignacio Llamas,et al.  Advections with Significantly Reduced Dissipation and Diffusion , 2007, IEEE Transactions on Visualization and Computer Graphics.

[10]  J. Benek,et al.  A flexible grid embedding technique with application to the Euler equations , 1983 .

[11]  T. Dupont,et al.  Back and forth error compensation and correction methods for removing errors induced by uneven gradients of the level set function , 2003 .

[12]  H. S. Tang,et al.  On Nonconservative Algorithms for Grid Interfaces , 1999, SIAM J. Numer. Anal..

[13]  James F. Epperson,et al.  An Introduction to Numerical Methods and Analysis , 2001 .

[14]  R. Courant,et al.  On the solution of nonlinear hyperbolic differential equations by finite differences , 1952 .

[15]  Xin Wang,et al.  Back and forth error compensation and correction method for linear hyperbolic systems with application to the Maxwell's equations , 2018, J. Comput. Phys. X.

[16]  Lili Hu.  Numerical algorithms based on the back and forth error compensation and correction , 2014 .

[17]  Yingjie Liu,et al.  Back and forth error compensation and correction methods for semi-lagrangian schemes with application to level set interface computations , 2006, Math. Comput..

[18]  Ignacio Llamas,et al.  FlowFixer: Using BFECC for Fluid Simulation , 2005, NPH.

[19]  P. Frederickson,et al.  Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction , 1990 .

[20]  Tao Tang,et al.  Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws , 2003, SIAM J. Numer. Anal..

[21]  Ronald Fedkiw,et al.  An Unconditionally Stable MacCormack Method , 2008, J. Sci. Comput..