Linearized and nonlinear numerical stability

for nonlinear PDEs

 

Cheng Wang
Associate Professor
Department of Mathematics
University of Massachusetts Dartmouth
North Dartmouth, MA 02747

E-Mail: cwang1@umassd.edu

Homepage: http://www.math.umassd.edu/~cwang/

 

Abstract: The theoretical issue of numerical stability and convergence analysis for a wide class of nonlinear PDEs is discussed in this talk. For most standard numerical schemes to certain nonlinear PDEs, such as the semi-implicit schemes for the viscous Burgers’ equation, a direct maximum norm analysis for the numerical solution is not available. In turn, a linearized stability analysis, based on an a-priori assumption for the numerical solution, has to be performed to make the local in time stability and convergence analysis go through. The linearized stability analysis usually requires a mild constraint between the time step and spatial grid sizes, therefore such a numerical stability is conditional. Instead, if a nonlinear numerical analysis could be directly derived, such as the convex splitting schemes for a class of gradient flows, a bound for the numerical solution becomes available, as a result of the energy stability. Therefore, the stability and convergence for these numerical schemes turn out to be unconditional. 

报告时间： 11:00- 12:00, July 24, 2017 (Monday)

报告地点： Room 316, the 3^rd floor of the environmental building, Renmin University of China

Research Interests of Dr. Cheng Wang

Dr. Cheng Wang’s research work is related to applied mathematics and scientific computing, especially the design and analysis of numerical methods for partial differential equations (PDEs). The equations aries from fluid mechanics, electro-magnetics, gradient flows, such as Cahn-Hilliard model in phase separation, phase field crystal (PFC) model in crystal growth, epitaxial thin film growth in material science, etc.

Dr. Cheng Wang is actively involved in many areas of research, including the following:

Fluid mechanics and geophysical flow

Numerical simulations of Maxwell equations in time domain

Gradient flows

Publications of Dr. Cheng Wang

1. Convergence of gauge method for incompressible flow. C. Wang, J.-G. Liu, Mathematics of Computation, 69, (2000), 1385-1407.

2. Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation. C. Wang, J.-G. Liu, Numerische Mathematik, 91, (2002), 543-576.

3. A fourth order scheme for incompressible Boussinesq equations. J.-G. Liu, C. Wang, H. Johnston, Journal of Scientific Computing, 18, (2003), 253-285.

4. Positivity property of second order flux-splitting schemes of compressible Euler equations. C. Wang, J.-G. Liu, Discrete and Continuous Dynamical Systems-Series B, 3, (2003), 201-228.

5. A fast finite difference method for solving Navier-Stokes equations on irregular domains. Z. Li, C. Wang, Communications in Mathematical Sciences, 1, (2003), 181-197.

6. Fourth order convergence of compact difference solver for incompressible flow. C. Wang, J.-G. Liu, Communications in Applied Analysis, 7, (2003), 171-191.

7. Surface pressure Poisson equation formulation of the primitive equations: Numerical schemes. R. Samelson, R. Temam, C. Wang, S. Wang, SIAM Journal of Numerical Analysis, 41, (2003), 1163-1194.

8. The primitive equations formulated in mean vorticity. C. Wang, Discrete and Continuous Dynamical Systems, Proceeding of ``International Conference on Dynamical Systems and Differential Equations``, 2003, 880-887.

9. High order finite difference methods for unsteady incompressible flows in multi-connected domains. J.-G. Liu, C. Wang, Computers and Fluids, 33, (2004), 223-255.

10. Analysis of a fourth order finite difference method for incompressible Boussinesq equations. C. Wang, J.-G. Liu, H. Johnston, Numerische Mathematik, 97, (2004), 555-594.

11. Convergence analysis of the numerical method for the primitive equations formulated in mean vorticity on a Cartesian grid. C. Wang, Discrete and Continuous Dynamical Systems-Series B, 4, (2004), 1143-1172.

12. Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow. M. Ghil, J.-G. Liu, C. Wang, S. Wang, Physica D, 197, (2004), 149-173.

13. Global weak solution of the planetary geostrophic equations with inviscid geostrophic balance J. Liu, R. Samelson, C. Wang, Applicable Analysis, 85, (2006), 593-606.

14. A fourth order numerical method for the planetary geostrophic equations with inviscid geostrophic balance. R. Samelson, R. Temam, C. Wang, S. Wang, Numerische Mathematik, 107, (2007), 669-705.

15. A fourth order numerical method for the primitive equations formulated in mean vorticity. with J.-G. Liu, C. Wang, Communications in Computational Physics, 4, (2008), 26-55.

16. A fourth order difference scheme for the Maxwell equations on Yee grid. A. Fathy, C. Wang, J. Wilson, S. Yang, Journal of Hyperbolic Differential Equations, 5, (2008), 613-642.

17. An accurate and stable fourth order finite difference time domain method. J. Wilson, C. Wang, S. Yang, A. Fathy, Y. Kang, IEEE Xplore, MTT-S International Microwave Symposium Digest, June 2008, 1369-1372.

18. A general stability condition for multi-stage vorticity boundary conditions in incompressible fluids. C. Wang, Methods and Applications of Analysis, 15, (2008), 469-476.

19. Structural stability and bifurcation for 2-D divergence-free vector with symmetry. C. Hsia, J.-G. Liu, C. Wang, Methods and Applications of Analysis, 15, (2008), 495-512.

20. Analysis of rapidly twisted hollow waveguides. J. Wilson, C. Wang, A. Fathy, Y. Kang, IEEE Transactions on Microwave Theory and Techniques, 57, (2009), 130-139.

21. Applications of twisted hollow waveguides as accelerating structures. J. Wilson, A. Fathy, Y. Kang, C. Wang, IEEE Transactions on Nuclear Science, 56, (2009), 1479-1486.

22. An energy stable and convergent finite-difference scheme for the phase field crystal equation. S. Wise, C. Wang, J. Lowengrub, SIAM Journal on Numerical Analysis, 47, (2009), 2269-2288.

23. Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation. Z. Hu, S. Wise, C. Wang, J. Lowengrub, Journal of Computational Physics, 228, (2009), 5323-5339.

24. Unconditionally stable schemes for equations of thin film epitaxy. C. Wang, X. Wang, S. Wise, Discrete and Continuous Dynamical Systems-Series A, 28, (2010), 405-423.

25. Global smooth solutions of the three-dimensional modified phase field crystal equation. C. Wang, S. Wise, Methods and Applications of Analysis, 17, (2010), 191-212.

26. An energy stable and convergent finite-difference scheme for the modified phase field crystal equation. C. Wang, S. Wise, SIAM Journal on Numerical Analysis, 49, (2011), 945-969.

27. Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: Application to thin film epitaxy. J. Shen, C. Wang, X. Wang, S. Wise, SIAM Journal on Numerical Analysis, 50, (2012), 105-125.

28. Long time stability of a classical efficient scheme for two dimensional Navier-Stokes equations. S. Gottlieb, F. Tone, C. Wang, X. Wang, D. Wirosoetisno, SIAM Journal on Numerical Analysis, 50, (2012), 126-150.

29. A linear energy stable scheme for a thin film model without slope selection. W. Chen, S. Conde, C. Wang, X. Wang, S. Wise, Journal of Scientific Computing, 52, (2012), 546-562.

30. Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers` equation. S. Gottlieb, C. Wang, Journal of Scientific Computing, 53, (2012), 102-128.

31. Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation. A. Baskaran, Z. Hu, J. Lowengrub, C. Wang, S. Wise, P. Zhou, Journal of Computational Physics, 250, (2013), 270-292.

32. Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation. A. Baskaran, J. Lowengrub, C. Wang, S. Wise, SIAM Journal on Numerical Analysis, 51, (2013), 2851-2873.

33. A linear iteration algorithm for a second order energy stable scheme for a thin film model without slope selection. W. Chen, C. Wang, X. Wang, S. Wise, Journal of Scientific Computing, 59, (2014), 574-601.

34. A local pressure boundary condition spectral collocation scheme for the three-dimensional Navier-Stokes equations. H. Johnston, C. Wang, J.-G. Liu, Journal of Scientific Computing, 60, (2014), 612-626.

35. Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations. Z. Guan, J. Lowengrub, C. Wang, S. Wise, Journal of Computational Physics, 277, (2014), 48-71.

36. A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation. Z. Guan, C. Wang, S. Wise, Numerische Mathematik, 128, (2014), 377-406.

37. A Fourier pseudospectral method for the ``Good" Boussinesq equation with second-order temporal accuracy. K. Cheng, W. Feng, S. Gottlieb, C. Wang, Numerical Methods for Partial Differential Equations, 31, (2015), 202-224.

38. An energy-conserving second order numerical scheme for nonlinear hyperbolic equation with an exponential nonlinear term. L. Wang, W. Chen, C. Wang, Journal of Computational and Applied Mathematics, 280, (2015), 347-366.

39. Simple finite element numerical simulation of incompressible flow over non-rectangular domains and the super-convergence analysis. Y. Xue, C. Wang, J.-G. Liu, Journal of Scientific Computing, 65 (2015), 1189-1216.

40. An $H^2$ convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn-Hilliard equation. J. Guo, C. Wang, S. Wise, X. Yue, Communications in Mathematical Sciences, 14 (2016), 489-515.

41. Convergence analysis of a fully discrete finite difference scheme for Cahn-Hilliard-Hele-Shaw equation. W. Chen, Y. Liu, C. Wang, S. Wise, Mathematics of Computation, 85 (2016), 2231-2257.

42. Global-in-time Gevrey regularity solution for a class of bistable gradient flows. N. Chen, C. Wang, S. Wise, Discrete and Continuous Dynamical Systems-Series B, 21 (2016), 1689-1711.

43. An energy stable, hexagonal finite difference scheme for the 2D phase field crystal amplitude equations. Z. Guan, V. Heinonen, J. Lowengrub, C. Wang, S. Wise, Journal of Computational Physics, 321 (2016), 1026-1054.

44. Stability and convergence of a second order mixed finite element method for the Cahn-Hilliard equation. A. Diegel, C. Wang, S. Wise, IMA Journal of Numerical Analysis, 36 (2016), 1867-1897.

45. Long time stability of high order multi-step numerical schemes for two-dimensional incompressible Navier-Stokes equations. K. Cheng, C. Wang, SIAM Journal on Numerical Analysis, 54 (2016), 3123-3144.

46. A second-order, weakly energy-stable pseudo-spectral scheme for the Cahn-Hilliard equation and its solution by the homogeneous linear iteration method. K. Cheng, C. Wang, S. Wise, X. Yue, Journal of Scientific Computing, 69 (2016), 1083-1114.

47. Preconditioned steepest descent methods for some regularized p-Laplacian problems. W. Feng, A. Salgado, C. Wang, S. Wise, Journal of Computational Physics, 334 (2017), 45-67.

48. Error analysis of a mixed finite element method for a Cahn-Hilliard-Hele-Shaw system. Y. Liu, W. Chen, C. Wang, S. Wise, Numerische Mathematik, (2016), accepted and published online.

49. Error analysis of an energy stable finite difference scheme for the epitaxial thin film growth model with slope selection with an improved convergence constant. Z. Qiao, C. Wang, S. Wise, Z. Zhang, International Journal of Numerical Analysis and Modeling, 14 (2017), 283-305.

50. Convergence analysis and error estimates for a second order accurate finite element method for the Cahn-Hilliard-Navier-Stokes system. A. Diegel, C. Wang, X. Wang, S. Wise, Numerische Mathematik, (2017), accepted and in press.

51. A second-order energy stable BDF numerical scheme for the Cahn-Hilliard equation. Y. Yan, W. Chen, C. Wang, S. Wise, Communications in Computational Physics, (2017), submitted and in review.

52. A second order operator splitting numerical scheme for the ``Good" Boussinesq equation. C. Zhang, H. Wang, J. Huang, C. Wang, X. Yue, Applied Numerical Mathematics, (2017), submitted and in review.

53. On the operator splitting and integral equation preconditioned deferred correction methods for the ``Good" Boussinesq equation. C. Zhang, J. Huang, C. Wang, X. Yue, Journal of Scientific Computing, (2017), submitted and in review.

54. A second order numerical scheme for nonlinear Maxwell`s equations using conforming finite element. C. Yao, Y. Lin, C. Wang, Journal of Computational Mathematics, (2017), submitted and in review.

55. A third order linearized BDF scheme for Maxwell`s equations with nonlinear conductivity using finite element method. C. Yao, C. Wang, Y. Kou, Y. Lin, International Journal of Numerical Analysis and Modeling, (2017), submitted and in review.

56. Convergence analysis for second order accurate convex splitting scheme for the periodic nonlocal Allen-Cahn and Cahn-Hilliard equations. Z. Guan, J. Lowengrub, C. Wang, Mathematical Methods in the Applied Sciences, (2017), submitted and in review.

57. An energy stable finite-difference scheme for functionalized Cahn-Hilliard equation and its convergence analysis. W. Feng, Z. Guan, J. Lowengrub, C. Wang, S. Wise, SIAM Journal on Numerical Analysis, (2017), submitted and in review.

58. A second order energy stable scheme for the Cahn-Hilliard-Hele-Shaw equation. W. Chen, W. Feng, Y. Liu, C. Wang, S. Wise, Discrete and Continuous Dynamical Systems-Series B, (2017), submitted and in review.

59. Convergence analysis and numerical implementation of a second order numerical scheme for the three-dimensional phase field crystal equation. L. Dong, W. Feng, C. Wang, S. Wise, Z. Zhang, Computers \& Mathematics with Applications, (2017), submitted and in review.

60. A refined truncation error estimate for long stencil fourth order finite difference approximation and its application to the Cahn-Hilliard equation. K. Cheng, W. Feng, C. Wang, S. Wise, (2017), in preparation.

61. Global-in-time Gevrey regularity solution to a three-dimensional Cahn-Hilliard-Stokes model. W. Chen, Y. Liu, C. Wang, S. Wise, (2017), in preparation.

62. The Strong-Stability-Preserving (SSP) scheme applied to the Integrating Factor (IF) form of Exponential Time Differencing (ETD) problems. S. Gottlieb, Z. Grant, C. Wang, (2017), in preparation.

63. An improved error analysis for a second-order convex splitting finite difference scheme for the Cahn-Hilliard equation. J. Guo, C. Wang, S. Wise, X. Yue, (2017), in preparation.

64. A spectral collocation method for two-dimensional incompressible fluid flows in vorticity formulation. H. Johnston, C. Wang, J.-G. Liu, (2017), in preparation.