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

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.