Page - 141 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4.3 OscillatingOne-DimensionalSystems 141 the combination of thesemethods yields an overall time-stepping scheme from tn to tnC1 that ismuchmoreaccurate than the individualstepswhichhaveerrorspro- portional to t and t2. This is indeed true: the numerical error goes in fact like C t4 foraconstantC,whichmeans that theerrorapproacheszeroveryquicklyas we reduce the timestep size, compared to theForwardEulermethod (error t), the Euler-Cromermethod (error t) or the 2nd-orderRunge-Kutta, orHeun’s, method(error t2). Note that the 4th-order Runge-Kuttamethod is fully explicit so there is never any need to solve linear or nonlinear algebraic equations, regardless of what f looks like.However, the stability is conditionalanddependsonf . There is a large familyof implicitRunge-Kuttamethods thatareunconditionallystable, but require solutionofalgebraicequations involvingf ateach timestep. TheOdespypackage has support for a lot of sophisticated explicit Runge-Kutta methods, but not yet implicitRunge-Kuttamethods. 4.3.8 MoreEffects:Damping,Nonlinearity,andExternalForces Ourmodelproblemu00C!2uD0 is thesimplestpossiblemathematicalmodel for oscillating systems. Nevertheless, thismodelmakes strongdemands to numerical methods, as we have seen, and is very useful as a benchmark for evaluating the performanceofnumericalmethods. Real-life applications involvemorephysical effects,which lead to adifferential equation with more terms and also more complicated terms. Typically, one has adampingforcef.u0/andaspring forces.u/. Both these forcesmaydependnon- linearlyon their argument,u0 oru. In addition, environmental forcesF.t/mayact on the system. For example, the classical pendulum has a nonlinear “spring” or restoring force s.u/ sin.u/, andair resistanceon thependulumleads to adamp- ing forcef.u0/ ju0ju0. Examplesonenvironmental forces includeshakingof the ground(e.g., due toanearthquake)aswell as forces fromwavesandwind. Withthreetypesofforcesonthesystem:F ,f ,ands, thesumofforcesiswritten F.t/ f.u0/ s.u/. Note theminus sign in front of f and s, which indicates that these functions are defined such that they represent forces acting against the motion. For example, springs attached to thewheels in a car are combinedwith effective dampers, each providing a damping forcef.u0/ D bu0 that acts against the springvelocityu0. Thecorrespondingphysical force is then f : bu0,which pointsdownwardswhenthespringisbeingstretched(andu0pointsupwards),while f actsupwardswhen the spring isbeingcompressed (andu0pointsdownwards). Figure 4.25 shows an example of amassm attached to a potentially nonlinear spring anddashpot, and subject to an environmental forceF.t/. Nevertheless, our generalmodelcanequallywellbeapendulumas inFig.4.26withs.u/Dmgsin and f.Pu/ D 1 2 CDA%P jP j (whereCD D 0:4,A is the cross sectional area of the body,and% is thedensityofair). Newton’s second lawfor the systemcanbewrittenwith themass timesacceler- ationon the left-handsideand the forceson the right-handside: mu00 DF.t/ f.u0/ s.u/: