Comparative Study of Matlab ODE Solvers for the Korakianitis and Shi Model

Abstract

Abstract. Changing parameters of the Korakianitis and Shi heart valve model over a cardiac cycle has led to the investigation of appropriate numerical technique(s) for good speed and accuracy. Two sets of parameters were selected for the numerical test. For the seven MATLAB ODE solvers, the computed results, computational cost and execution time were observed for varied error tolerance and initial time steps. The results were evaluated with descriptive statistics; the Pearson correlation and ANOVA at 0.05 . The dependence of the computed result, accuracy of the method, computational cost and execution time of all the solvers, on relative tolerance and initial time steps were ascertained. Our findings provide important information that can be useful for selecting a MATLAB ODE solver suitable for differential equation with time varying parameters and changing stiffness properties. IntroductionThe Korakianitis and Shi (KS) heart valve model is a zero-dimensional, second order, nonlinear, Ordinary Differential Equation (ODE) that describes the dynamics of the human heart valve. The equation is embedded in the overall model of other parts of the entire cardiovascular system [1]. The equation may change from being non-stiff to stiff and vice versa, as its parameter changes within a heart cycle. An equation is said to be stiff if certain numerical schemes that are not absolutely stable cannot be used to seek an approximate solution to the problem [2]. Consequently, the conventional fourth order explicit Runge-Kutta method used to find a solution to the KS heart valve model in previous studies [1,[3][4] may not give accurate results for large step size, while smaller steps size may increase the round-off errors and increase computation time.On cannot exhaust the list of numerical techniques and their suitability for solving ODE as older ones are being modified and newer ones are developed. Butcher [5] reported a good number of these solvers developed over the 20 th century. Evaluations of ODE solvers have a long history [6-9] and earlier intentions were either to ascertain their effectiveness [6][7] or to categorize them as either stiff or non-stiff solvers [10][11][12][13]. Some researchers, however, reported the performance of the solving environment for some of these ODE solvers [14][15][16]. Very recently, focus has been to investigate ODE solvers for specific problem sets [17][18][19][20].The MATLAB software is widely used in engineering studies. Its ODE suite has seven powerful solvers, three of which are explicit schemes while the other four are implicit schemes. The ode45, ode23 and ode113 are the three explicit schemes while ode23t, ode23s, ode15s and ode23tb are the four implicit solvers. Interested readers are referred to the literatures [21-23] for mathematical details of the different solvers in the MATLAB ODE suite.Runge-Kutta methods are widely discussed in the literature [24][25][26][27]. The ode23 and ode45 solvers implement "Runge-Kutta formula of order 2 and 3" [21] and the "Dormand-Prince (4,5) pair"...

Development and validation of a simplified nonlinear dynamic model of a passive twin-tube hydraulic shock absorber

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Fixed Points and Stability Analysis in the Motion of Human Heart Valve Leaflet

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