Accuracy Tests on Built-In Algorithms Applied to the Lorenz System

This work investigate, An idea of checking accuracy of algorithms from mathematical black box by means of residual functions. Lorenz system is used as case study as the chaotic system does not have analytical solution. The numerical procedures examined include BDF, Adams method and Implicit Runge Kutta methods. The interval of numerical results is t ∈ [0; 10].

VARIATIONAL MUTI-STEPS METHOD TO SOLVE DAMPED OSCILLATION EQUATION

<p class="abstrak"><span lang="IN">This paper aims to identifying the numerical method accuracy of the analytical solution of the damped oscillation equation motion. Adams method of 4th order, Milne method and Adams-Simpson method are used to find numerical solutions. Value of <em>y(1)</em>, <em>y(2)</em>, <em>y(3)</em> obtained from The 4th order Runge-Kutta method. They used as initial value of multistep method. Then, the numerical solution result was compared with analytical solution. From the research result, it is found that 4th order Adams method has the best accuracy.</span></p>

TLISMNI/Adams algorithm for the solution of the differential/algebraic equations of constrained dynamical systems

This paper examines the performance of the 3rd and 4th order implicit Adams methods in the framework of the two-loop implicit sparse matrix numerical integration method in solving the differential/algebraic equations of heavily constrained dynamical systems. The variable-step size two-loop implicit sparse matrix numerical integration/Adams method proposed in this investigation avoids numerical force differentiation, ensures satisfying the nonlinear algebraic constraint equations at the position, velocity, and acceleration levels, and allows using sparse matrix techniques for efficiently solving the dynamical equations. The iterative outer loop of the two-loop implicit sparse matrix numerical integration/Adams method is aimed at achieving the convergence of the implicit integration formulae used to solve the independent differential equations of motion, while the inner loop is used to ensure the convergence of the iterative procedure used to satisfy the algebraic constraint equations. To solve the independent differential equations, two different implicit Adams integration formulae are examined in this investigation; a 3rd order implicit Adams-Moulton formula with a 2nd order explicit predictor Adams Bashforth formula, and a 4th order implicit Adams-Moulton formula with a 3rd order explicit predictor Adams Bashforth formula. A standard Newton–Raphson algorithm is used to satisfy the nonlinear algebraic constraint equations at the position level. The constraint equations at the velocity and acceleration levels are linear, and therefore, there is no need for an iterative procedure to solve for the dependent velocities and accelerations. The algorithm used for the error check and step-size change is described. The performance of the two-loop implicit sparse matrix numerical integration/Adams algorithm developed in this investigation is evaluated by comparison with the explicit predictor-corrector Adams method which has a variable-order and variable-step size. Simple and heavily constrained dynamical systems are used to evaluate the accuracy, robustness, damping characteristics, and effect of the outer-loop iterations of the proposed implicit schemes. The results obtained in this investigation show that the two-loop implicit sparse matrix numerical integration methods proposed in this study can be more efficient for stiff systems because of their ability to damp out high-frequency oscillations. Explicit integration methods, on the other hand, can be more efficient in the case of non-stiff systems.