Strong-Stability-Preserving, K-Step, 5- to 10-Stage, Hermite-Birkhoff Time-Discretizations of Order 12

Truong Nguyen-Ba, Huong Nguyen-Thu, Re´mi Vaillancourt

.

DOI: 10.4236/ajcm.2011.12008                

Abstract

We construct optimal k-step, 5- to 10-stage, explicit, strong-stability-preserving Hermite-Birkhoff (SSP HB) methods of order 12 with nonnegative coefficients by combining linear k-step methods of order 9 with 5- to 10-stage Runge-Kutta (RK) methods of order 4. Since these methods maintain the monotonicity property, they are well suited for solving hyperbolic PDEs by the method of lines after a spatial discretization. It is seen that the 8-step 7-stage HB methods have largest effective SSP coefficient among the HB methods of order 12 on hand. On Burgers’ equations, some of the new HB methods have larger maximum effective CFL numbers than Huang’s 7-step hybrid method of order 7, thus allowing larger step size.

Keywords

Strong Stability Preserving, Hermite-Birkhoff Method, SSP Coefficient, Time Discretization, Method of Lines, Comparison with Other SSP Methods

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. Harten, “High Resolution Schemes for Hyperbolic Conser-vation Laws,” Journal of Computational Physics Vol. 49, No. 3, 1983, pp. 357-393. doi:10.1016/0021-9991(83)90136-5
[2]	S. Osher and S. Chakravarthy, “High Resolution Schemes and the Entropy Condition,” SIAM Journal on Numerical Analysis, Vol. 21, No. 5, 1984, pp. 955-984. doi:10.1137/0721060
[3]	P. K. Sweby, “High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws,” SIAM Journal on Numerical Analysis, Vol. 21, No. 5, 1984, pp. 995-1011. doi:10.1137/0721062
[4]	B. Cockburn and C. W. Shu, “TVB Runge-Kutta Local Projection Discon-tinuous Galerkin Finite Element Method for Conservation Laws II: General Framework,” Mathematics of Computation, Vol. 52, No. 186, 1989, pp. 411-435.
[5]	S. Gottlieb, D. I. Ketcheson and C. W. Shu, “High Order Strong Stability Pre-serving Time Discretization,” Journal of Scientific Computing, Vol. 38, No. 3, 2009, pp. 251-289.
[6]	C. Huang, “Strong Stability Preserving Hybrid Methods,” Applied Numerical Mathematics, Vol. 59, No. 5, 2009, pp. 891-904. doi:10.1016/j.apnum.2008.03.030
[7]	T. Nguyen-Ba, E. Kengne and R. Vaillancourt, “One-Step 4-Stage Her-mite-Birkhoff-Taylor ODE Solver of Order 12,” The Canadian Applied Mathematics Quarterly, Vol. 16, No. 1, 2008, pp. 77-94.
[8]	S. Gottlieb, C. W. Shu and E. Tadmor, “Strong Stability- Preserving Highorder Time Discretization Methods,” SIAM Review, Vol. 43, No. 1, 2001, pp. 8-112 doi:10.1137/S003614450036757X
[9]	S. J. Ruuth and R. J. Spiteri, “High-Order Strong-Stability-Preserving Runge-Kutta Methods with Down-Biased Spatial Discretizations,” SIAM Journal on Numerical Analysis, Vol. 42, No. 3, 2004, pp. 974-996. doi:10.1137/S0036142902419284
[10]	S. Gottlieb, “On High Order Strong Stability Preserving Runge-Kuttaand Multi Step Time Discretizations,” Journal of Scientific Computing, Vol. 25, No. 1-2, 2005, pp. 105-128.
[11]	C. W. Shu and S. Osher, “Efficient Implementation of Essentially Nonoscillatory Shock-Capturing Schemes,” Journal of Scientific Computing, Vol. 77, No. 2, 1988, pp. 439-471. doi:10.1016/0021-9991(88)90177-5
[12]	C. W. Shu, “To-tal-Variation-Diminishing Time Discretizations,” SIAM Jour-nal on Numerical Analysis, Vol. 9, No. 6, 1988, pp. 1073-1084. doi:10.1137/0909073
[13]	S. Gottlieb and C. W. Shu, “Total Variation Diminishing Runge-Kutta Schemes,” Mathematics of Computation, Vol. 67, No. 221, 1998, pp. 73-85. doi:10.1090/S0025-5718-98-00913-2
[14]	R. J. Spiteri and S. J. Ruth, “A New Class of Optimal High-Order Strong-Stability-Preserving Time-Stepping Schemes,” SIAM Journal on Numerical Analysis, Vol. 40, No. 2, 2002, pp. 469-491. doi:10.1137/S0036142901389025
[15]	R. J. Spiteri, S. J. Ruuh, “Nolinear Evoluton Using Optimal Fourth-Order Strong-Stability-Preserving Runge- Kutta Methods, Journal of Mathematics and Computers in Simulation, Vol. 62, No. 1-2, 2003, pp. 125-135. doi:10.1016/S0378-4754(02)00179-9
[16]	S. J. Ruuth and R. J. Piteri, “Two Barriers on Strong- Stability-Preserving Time Discretization Methods,” Journal on Scientific Computing, Vol. 17, No. 1-4, 2002, pp. 211-220. doi:10.1023/A:1015156832269
[17]	S. J. Ruuth, “Global Op-timization of Explicit Strong- Stability-Preserving Runge-Kutta Methods,” Mathematics of Computation, Vol. 75, No. 253, 2006, pp. 183-207. doi:10.1090/S0025-5718-05-01772-2
[18]	W. Hundsdorfer, S. J. Ruuth and R. J. Spiteri, “Monotonicity Preserving Linear Multistep Methods,” SIAM Journal on Numerical Analysis, Vol. 41, No. , 2003, pp. 605-623. doi:10.1137/S0036142902406326
[19]	I. Higueras, “On Strong Stability Preserving Methods,” Journal of Scientific Computing, Vol. 21, No. , 2004, pp. 193-223. doi:10.1023/B:JOMP.0000030075.59237.61
[20]	S. J. Ruuth and W. Hundsdorfer, “High-Order Linear Multistep Methods with General Monotonicity and Boundedness Properties,” Journal of Computational Physics, Vol. 209, No. 1, 2005, pp. 226-248. doi:10.1016/j.jcp.2005.02.029
[21]	G. Jiang and C. W. Shu, “Efficient Implementation of Weighted ENO Schemes,” Jour-nal of Computational Physics, Vol. 126, No. 1, 1996, pp. 202-228. doi:10.1006/jcph.1996.0130
[22]	C. Laney, “Computational Gasdynamics,” Cambridge University Press, Cambridge, 1998. doi:10.1017/CBO9780511605604