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.


Introduction
We are concerned with the numerical solution of initial value problems d ( , ( )) d y f t y t t  , 0 0 ( ) y t y  . ( where the function f is such that for all 0 t   .Here  may be a norm or, more generally, any convex functional.It is also assumed that f satisfies the discrete analog of (2), ( , ) for the forward Euler method.Here n y is a numerical approximation to 0 ( ) y t n t   .We are interested in higher-order accurate multistep HB methods that preserve the monotonicity property for max 0 FE t t c t       whenever condition (3) holds.Here k represents the number of previous steps used to compute the next solution value and c , called the SSP coefficient, depends only on the numerical integration method but not on f .The monotonicity property (4) is desirable as it mimics property (2) of the true solu-tion and prevents growth of errors.
Strong-stability-preserving (SSP) methods have been developed to satisfy the monotonicity property (4) for system (1) whenever condition (3)  The main application of such monotonicity results are found in the numerical solution of hyperbolic PDEs, in particular, of conservation laws.For the one-dimensional equation the spatial derivative ( ) x g y can be approximated by a conservative finite difference or finite element at , j x j 1, 2, , N   , (see, for example, [1][2][3][4]).This spatialsemidiscretization will lead to system (1) of ODEs.
In this paper, to solve system (1), we construct new explicit, multistep, multistage, SSP general linear timediscretization methods of order 12 with nonnegative coefficients.These methods, which we call SSP Hermite-Birkhoff (SSP HB), because their construction involves HB interpolation polynomials (see Section 2), are combinations of linear k -step methods of order 9 and s -stage RK methods of order 4. The objective of high-order SSP HB time discretizations is to maintain the 1 monotonicity property (4) while achieving higher-order accuracy in time, perhaps with a modified CFL restriction, measured here with an SSP coefficient, ( ) c HBks : The SSP coefficient, historically called CFL coefficient, describes the ratio of the strong-stability-preserving time step to the strongly-stable forward Euler time step (see [5]).Since our arguments are based on convex decompositions of high-order methods in terms the SSP FE method, such high-order methods preserve SSP in any norm once FE is shown to be strongly stable.
Several new explicit 6-to 10-stage SSP HB methods with nonnegative coefficients presented here have been found by computer search.
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 [6], thus allowing larger step size.
In particular, no counterparts of k -step HB methods of order 12 have been found in the literature among hybrid and general linear multistep methods.Moreover, the 8-step, 7-stage HB method has largest effective SSP coefficient among the 12th-order HB methods on hand.
Section 2 introduces 5-to 10-stage SSP HB methods.Order conditions are listed in Section 3. Section 4 derives the Shu-Osher representation of k -step 5-to 10-stage HB methods of order 12. New SSP HB methods are formulated as solutions of optimization problems in Section 5. Section 6 compares the effective SSP coefficients of different methods and lists several new SSP HB methods.Numerical results for several methods applied to Burgers' equations are presented in Section 7. Appendix A lists the Shu-Osher representation of some of the best HBks methods considered in this paper.

K -step, S -stage SSP HB Methods of Order 12
Notation 1: The following notation will be used: • k denotes the number of steps of a given method, • s denotes the number of stages of a given method, • HBks denotes k -step, s -stage SSP Hermite-Birkhoff methods of order 12, • HMk denotes k -step SSP hybrid methods of order 7.
All HBks methods considered in this work are SSP and of order 12 unless specified otherwise.Therefore the denominations "SSP" and "order 12" will often be omitted in what follows.
Notation 2: The abscissa vector An HBks method requires the following s formulae to perform integration from n t to 1 n t  , where, for simplicity, 1 0 c  is used in the summations.By con-

Y y
 .An HB polynomial of degree 2 3 k i   is used as predictor i P to obtain the th i stage value i Y to order 9, An HB polynomial of degree 2 2 k s   is used as integration formula to obtain 1 n y  to order 12:

Order Conditions of HBks
To derive the order conditions for HBks we shall use the following expressions coming from the backsteps of the methods: , .
As in the construction of RK methods, we impose the following simplifying conditions on the abscissa vector   , , , , Forcing an expansion of the numerical solution produced by formulae ( 7)-( 8) to agree with a Taylor expansion of the true solution, we obtain multistep-and RK-type order conditions that must be satisfied by HBks .To reduce the large number of RK-type order conditions, we impose the following simplifying assumptions, as in similar searches for ODE solvers [7]: Note that (11) with 0 k  reduces to (10).There are seven sets of equations to be solved: where the backstep parts, ( ) B j , are defined by ! 1, 2, ,12.

Shu-Osher Representation of HBks
We rewrite HBks in the Shu-Osher representation asconvex combinations of FE to show that they satisfy SSP conditions.Firstly, if we let then formulae (7) and ( 8) become Replacing the index i by m in formula (7), we express n y as a function of m Y ,  22) as (25) in the Shu-Osher equivalent form: where the coefficients are , , Thirdly, the representation (24,25), under the assumptions that all coefficients are nonnegative, implies that the HBkp are SSP.In fact, one finds that the following functions are convex combinations of forward Euler steps: • In (24) for 2, 3, , i s   , the first and second bracketed terms are sums of FE steps with step sizes , and , respectively.
• In (25), the first and second bracketed terms are sums of FE steps with step sizes and , respectively.
One can easily verify that Provided all the coefficients ij A , ij e , j A , j e are nonnegative, the following straightforward extension of a result presented in [6,8]  where the SSP coefficient ( ) c HBks is the minimum of the four numbers: with the convention that / 0    , under the assumption that all coefficients of (24) -(25) are nonnegative.

Construction of Optimal HBks
Since HBks contain many free parameters when k is sufficiently large, we use the Matlab Optimization Toolbox to search for the methods with largest ( ) c HBks for different k and s .To optimize HBks , we maximize ( ) c HBks of Theorem 1 by solving the nonlinear programming problem , , , where all the numbers in all pairs   , , 0, 1, , 1 , are nonnegative.Null pairs, {0, 0}, are not included in the minimization process if they occur.Besides the nonnegativity constraints on all variables, the objective func-tion ( 27) is subject to • the convex combinations constraints (20), • the simplifying assumptions ( 10) and ( 11) for HBks , • the order conditions ( 12) to (18) for HBks , • the conditions on the abscissa vector:

Comparing Effective SSP Coefficients
Definition 1: (See [9]) The effective SSP coefficients of an SSP method M is denoted by where l is the number of function evaluations of method M per time step and ( ) The SSP coefficients, ( ) c HM , of hybrid methods are defined in [6].In this paper, , provide a fair comparison between methods of the same order, although, in practice, starting methods and storage issues may also be important.Gottlieb [10] pointed out that one looks for highorder SSP methods M with ( ) c M as large as possible, taking their computational costs and orders into account.
We briefly review the developments of SSP methods.Shu and Osher [11] constructed a series of second-to fifth-order SSP RK methods, several of which are downwinded ones.Shu [12] found a class of first-order SSP RK methods with very large SSP coefficients, as well as one-to five-step SSP methods of orders two to five.Gottlieb and Shu [13] derived optimal s -stage SSP RK methods of order s for 2, 3 s  , and proved that for 4 s  there is no such SSP method with nonnegative coefficients.Spiteri and Ruuth [14,15] studied optimal s -stage SSP RK methods of order p with s p  for 4 p  .They proved the nonexistence of fifth order SSP RK methods with nonnegative coefficients [16] and constructed some fifth-order methods of seven to nine stages with downwind-biased spatial discretization [9].A 10stage method of order 5 was given in [17].Hunds-dorfer, Ruuth and Spiteri [18] proved that the implicit Euler method can unconditionally preserve the strong stability of the FE method (see also [19]) and studied multistep methods with specific starting procedures.
Ruuth and Hundsdorfer [20] pointed out that linear multistep methods of order five require at least seven steps.Huang [6]

Numerical Results
From now on, we use the total variation semi-norm, , and say that a method is total variation diminishing We compare our new methods numerically with 7 HM of Huang.

Starting Procedure
To maintain the TVD property (31), the necessary starting values for HBkp were obtained by RK54 with small initial step size, 1 1.0 0.4 h e   (approximatively).

Comparing HBks with 7 HM with a Unit Downstep Initial Condition
As a first comparison of HBks of order 12 with Huang's 7-step 7 HM of order 7 [6], we consider Burgers' equation in Problem 1.
Problem 1: Burgers' equation with a unit downstep initial condition: and boundary condition ( 1, )  .This leads to the semi-discrete system where ( , ) (1/ 2) j f  is the numerical flux, which typically is a Lipschitz continuous function of several neighboring values ( ) j u t (see [21] for details).A time discretization can then be applied to (33).
We consider the total variation norm of the numerical solution at 1.8 . For this purpose, we let eff n be the largest effective CFL number defined as such that the TV error in the numerical solution satisfies the inequality , as a function of s for Problem 1 are listed in Table 3.

Comparing HBks and 7 HM with a Square-Wave Initial Condition
As a second comparison, we consider Burgers' equation with a square-wave initial value in Problem 2, which is test case 4 of Laney's five test problems [22, p.312].
Problem 2: Burgers' equation with a square wave initial condition:

Conclusions
New optimal explicit k -step, s -stage ( 5, 6,...,10 s  ) SSP Hermite-Birkhoff methods, HBks , of orders 12 with nonnegative coefficients are constructed by combining linear k -step methods of order 9 with 5-to 10-stage Runge-Kutta methods of order 4. No counterparts of HBks of order 12 have been found in the literature among hybrid and general linear multistep methods.
Moreover, the 8-step 7-stage 87 HB has largest effective SSP coefficient among the 12th-order HB methods on hand.It is found that some of new HBks have larger effective SSP coefficients and larger maximum effective CFL numbers than Huang's 7-step hybrid method of order 7 on Burgers' equations.

Appendix
This appendix lists the Shu-Osher representation of some of the best HBks methods considered in this paper with large ( ) c HBks , eff ( ) c HBks and abscissa vector   HBks methods (24,25) are SSP provided holds.Theorem 1: If the forward Euler method FE is SSP under the CFL condition FE t t    , then the k -step, s -stage It was also observed numerically that the TVD property (31) holds with error (35) for the methods listed in

ff ff PEG n HBks n HM for HBks and 7 HM , and ratio
n R HM  .