GENERALIZED HIGH ORDER SECOND DERIVATIVE EXTENDED BOUNDARY VALUE METHODS FOR STIFF SYSTEMS OF ODES

ABSTRACT

The difficulty in solving stiff systems increases either when the stiffness ratio becomes very large or when the constant multiplying the highest derivative of the boundary value problem becomes very small or when the system is both stiff and oscillatory or when the stiff system have its large eigenvalues of the Jacobian matrix close to the imaginary axis. Thus, a suitable method to handle with all cases has been proven to have the quality of being of high order, A-stable and 0-stable with considerably small error constant. In this study, highly stable Boundary Value Methods (BVMs) suitable for stiff systems with any of the above mentioned properties are introduced. The approach adopted in this study is that of solving the continuous Initial Value Problem (IVP) and Boundary Value Problem (BVP) by means of a discrete boundary value problem. This approach is generally referred to as BVM approach and methods developed in this way are referred to as BVMs. The BVM families of methods introduced in this study are derived using the Taylors series expansion procedures. Four new classes of Cash-type second derivative extended backward differentiation formulas are developed and implemented as BVMs in this study. They are 𝐴𝑘1,𝑘2 −stable and 𝑂𝑘1,𝑘2 −stable with (𝑘1, 𝑘2 ) −boundary conditions for values of the step number 𝑘 ≥ 1 and order up to 𝑘 + 6 with no barriers as concerning the maximum attainable order. The BVM approach of implementation considered circumvents the accumulation of various local truncation errors inherent in the usual predictor-corrector approach. Their suitability for linear and nonlinear stiff systems are tested on some standard stiff initial value problems and stiff boundary value problems. The new schemes are found to be highly competitive with existing methods.