Accelerating the rate of convergence of some efficient schemes for two-stage Gauss method

Abstract

The non-linear equations obtaining from the implicit s – stage Runge-Kutta methods have been solved by various iteration schemes. A scheme has been developed, which is computationally more efficient and avoids expensive vector transformations. The rate of convergence of this scheme is examined when it is applied to the scalar test differential equation 􀀒􀀢 = 􀀣􀀒 and the convergence rate depends on the spectral radius 􀀤[􀀘(􀀦)] of the iteration matrix 􀀘(􀀦), where 􀀦 = ℎ􀀣 and ℎ is the step-size. In this scheme, supremum of a lower bound for 􀀤[􀀘(􀀦)] is minimized over the left half 􀀦- plane with the constraints requiring super-linear convergence at 􀀦 = 0 and 􀀦 → ∞ .Two new schemes with parameters are obtained for the two-stage Gauss-method. Numerical experiments are carried out in order to evaluate and compare the efficiency of the new schemes and the original scheme. Consider an initial value problem for stiff system of ordinary differential equations 􀀒􀀢 = 􀀬􀀭􀀒(􀀟)􀀮, 􀀒(􀀯) = 􀀰, 􀀬: ℝ􀀐 → ℝ􀀐. An s-stage implicit Runge-Kutta method computes an approximation 􀀒􀀳􀀴􀀜 to the solution x (􀀟􀀳􀀴􀀜) at discrete point 􀀟􀀳􀀴􀀜 = 􀀟􀀳 + ℎ by 􀀒􀀳􀀴􀀜 = 􀀒􀀳 + ℎ Σ 􀀛􀀶 􀀷 􀀶 􀀸􀀜 􀀬(􀀔􀀶 ), where 􀀔􀀜 ,􀀔􀀙 ,…,􀀔􀀷, satisfy sn equations 􀀔􀀶 􀀸 􀀒􀀳 + ℎ 􀀺 􀀯􀀶􀀻 􀀷 􀀶􀀸􀀜 􀀬􀀭􀀔􀀻􀀮, ), 􀀼 = 1,2, . . . , 􀁀. 􀁁 = 􀁂􀀯􀀶􀀻􀁃 is the real coefficient matrix and 􀀛 = [􀀛􀀜 ,􀀛􀀙 ,…,􀀛􀀷]􀁄 is the column vector of the Runge-Kutta method. Let 􀀔 = 􀀔􀀜 ⊕ 􀀔􀀙 ⊕ … ⊕ 􀀔􀀷 ∈ ℝ􀀷􀀐and 􀁇(􀁈) = 􀀬(􀀔􀀜) ⊕ 􀀬(􀀔􀀙) ⊕ … ⊕ 􀀬(􀀔􀀷) ∈ ℝ􀀷􀀐. Then the above equation in 􀀔􀀜 ,􀀔􀀙 ,…,􀀔􀀷 may be written by 􀁈 = 􀁉 ⊗ 􀀒􀀳 + ℎ(􀁁 ⊗ 􀁋􀀐)􀁇(􀁈), where 􀁉 = (1,1, … ,1)􀁄 and (􀁁 ⊗ 􀁋􀀐) is the tensor product of the matrix 􀁁 with 􀀑 × 􀀑 identity matrix 􀁋􀀐. The efficient scheme, which has been already proposed, is given by [􀁋􀀷 ⊗ (􀁋􀀐 − ℎ􀁌􀁍)]􀁎􀁏 = (􀁐 ⊗ 􀁋􀀐)(􀁉 ⊗ 􀁑􀀳 – 􀁈􀁏) + (􀁓 ⊗ 􀁋􀀐)( 􀁉 ⊗ 􀁑􀀳 – 􀁈􀁏􀁔􀀜) + ℎ(􀁕 ⊗ 􀁋􀀐)􀁇(􀁈􀁏) + ℎ(􀀌 ⊗ 􀁋􀀐)􀁇(􀁈􀁏􀁔􀀜), 􀁖 = 1,2, …, In this scheme, supremum of a lower bound for 􀀤[􀀘(􀀦)] is minimized over ℂ􀁔, where ℂ􀁔 = {􀀦 ∈ 􀁙 /􀀌􀁉 (􀀦) ≤ 0 } with the constraints 􀀤[􀀘(􀀦)] = 0 at 􀀦 = 0 and 􀀤[􀀘(􀀦)] = 0 at 􀀦 → ∞. The parameters for the two-stage Gauss method are obtained and Numerical experiments are carried out.