Variable step variable order block backward differentiation formulae for solving stiff ordinary differential equations

Block Backward Differentiation Formulae (BBDF) method with variable step variable order approach (VSVO) for solving stiff Ordinary Differential Equations (ODEs) is described in this thesis. The research on Variable Step Variable Order Block Backward Differentiation Formulae (VSVO-BBDF) method is divided into two parts where the first part attempts to solve first order stiff ODEs, whereby second order stiff ODEs are considered subsequently. Initially, the computation of Dth-order variable step BBDF (VS-BBDF) method of order three up to five is presented. The detailed algorithms of VSVO-BBDF method is discussed to show the crucial parts of the order and stepsize selections. Prior to getting the numerical results, the MATLAB’s suite of ODEs solvers namely ode15s and ode23s is applied for the numerical comparison purposes. Meanwhile, the consistency and zero stability properties that lead to the convergence of the method are also discussed. Finally, the implementation of the VSVO-BBDF(2) method for the solution of second order stiff ODEs is analyzed. The derivation of the method of order two up to four, as well as the strategies in choosing the order and stepsize are elaborated. Similarly, numerical results are obtained after a fair comparison is made between VSVO-BBDF(2) and stiff ODEs solvers in MATLAB. In conclusion, the results display positive trends in reducing the total number of steps and increasing the accuracy of the approximations. The results also show that VSVO-BBDF method reduces the time execution for solving first and second order stiff ODEs as compared to MATLAB’s ODEs solvers. Therefore, these methods serve the purpose of significant alternatives for solving stiff ODEs.

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