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  2. Numerov's method - Wikipedia

    en.wikipedia.org/wiki/Numerov's_method

    Numerov's method (also called Cowell's method) is a numerical method to solve ordinary differential equations of second order in which the first-order term does not appear. It is a fourth-order linear multistep method. The method is implicit, but can be made explicit if the differential equation is linear.

  3. Numerical methods for ordinary differential equations - Wikipedia

    en.wikipedia.org/wiki/Numerical_methods_for...

    For example, the second-order equation y′′ = −y can be rewritten as two first-order equations: y′ = z and z′ = −y. In this section, we describe numerical methods for IVPs, and remark that boundary value problems (BVPs) require a different set of tools. In a BVP, one defines values, or components of the solution y at more than one ...

  4. Settling time - Wikipedia

    en.wikipedia.org/wiki/Settling_time

    Settling time depends on the system response and natural frequency. The settling time for a second order , underdamped system responding to a step response can be approximated if the damping ratio ζ ≪ 1 {\displaystyle \zeta \ll 1} by T s = − ln ⁡ ( tolerance fraction ) damping ratio × natural freq {\displaystyle T_{s}=-{\frac {\ln ...

  5. Runge–Kutta methods - Wikipedia

    en.wikipedia.org/wiki/Runge–Kutta_methods

    The Gauss–Legendre method with s stages has order 2s, so its stability function is the Padé approximant with m = n = s. It follows that the method is A-stable. [34] This shows that A-stable Runge–Kutta can have arbitrarily high order. In contrast, the order of A-stable linear multistep methods cannot exceed two. [35]

  6. Root locus analysis - Wikipedia

    en.wikipedia.org/wiki/Root_locus_analysis

    The definition of the damping ratio and natural frequency presumes that the overall feedback system is well approximated by a second order system; i.e. the system has a dominant pair of poles. This is often not the case, so it is good practice to simulate the final design to check if the project goals are satisfied.

  7. Newmark-beta method - Wikipedia

    en.wikipedia.org/wiki/Newmark-beta_method

    It is widely used in numerical evaluation of the dynamic response of structures and solids such as in finite element analysis to model dynamic systems. The method is named after Nathan M. Newmark , [ 1 ] former Professor of Civil Engineering at the University of Illinois at Urbana–Champaign , who developed it in 1959 for use in structural ...

  8. Symplectic integrator - Wikipedia

    en.wikipedia.org/wiki/Symplectic_integrator

    The Verlet method is the second-order integrator with = and coefficients =, =, = =. Since c 1 = 0 {\displaystyle c_{1}=0} , the algorithm above is symmetric in time. There are 3 steps to the algorithm, and step 1 and 3 are exactly the same, so the positive time version can be used for negative time.

  9. Double integrator - Wikipedia

    en.wikipedia.org/wiki/Double_integrator

    Feedback system with a PD controller and a double integrator plant In systems and control theory , the double integrator is a canonical example of a second-order control system. [ 1 ] It models the dynamics of a simple mass in one-dimensional space under the effect of a time-varying force input u {\displaystyle {\textbf {u}}} .