Runge Kutta Behavior in the Presence of a Jump Discontinuity

Transcription

1 Runge Kutta Behavior in the Presence of a Jump Discontinuity Allen Back July, 5 The graphs below appear to show motion at constant speed v = 4 along a straight line They were generated by numerical integration of a second order system of differential equations But the slope of the straight line x = x(t) is not the constant v! In fact the apparent velocity is v 6 Velocity vs Time Position vs Time v x time time The above system arose from numerical integration of v = f(v) x = v

2 where f(v) = b sgn(v) and b = By sgn(v) we mean if v > sgn(v) = if v = if v < The numerical integration method was fourth order Runge-Kutta with a step size of and an initial value of (v, x) = (4, ) The Runge Kutta method (eg []) for the two dimensional system (v, x ) = F (t, v, x) is derived under the assumption that F is three times differentiable Here we see that the right hand side of the equation is not even a continuous function of v So it is not surprising that Runge Kutta fails to give an accurate solution to this system What is striking is the manner of its failure The following result shows that the behavior with the above initial condition always happens for initial values of v close enough to Theorem A: Let b > and consider the system of ODE s (v, x ) = ( b sgn(v), v) with initial condition (v(t ), x(t )) = (v, x ) Then for any step size h > with < v < bh, one step of fourth order Runge Kutta leaves v unchanged at v and x changed from x to ( x + h v sgn(v ) bh ) 6 Since numerical integration preserves the condition v < bh, we can repeat as many steps as we like resulting in a numerical solution with constant velocity (v value) v and constant apparent velocity (rate of change of x) of (v bh 6 ) Intuitively, Runge Kutta involves making four preliminary estimates of where the next step ends up, each making use of what has been learned thusfar We will refer to the process of obtaining the next preliminary estimate as a substep and use the superscripts st, nd, rd, or 4th to indicate which of the four preliminary estimates is being reported We finalize the step by taking a suitable weighted average

3 In referring to the details of the Runge-Kutta compution, we will also use the subscripts IN, MID, or NXT These indicate which time value (t, t + h, or t + h respectively) a y value pertains to In vector form for the system y = F (t, y) with initial condition y(t ) = y IN, the Runge-Kutta method may be expressed precisely as follows: Substep Value of F st F st = F (t, y IN ) nd F nd = F (t + h, yst rd F rd = F (t + h, ynd 4th F 4th = F (t + h, y rd MID ) MID ) NXT ) Substep Estimate of y(t + h ) Estimate of y(t + h) st y MID st = y IN + h F st y NXT st = y IN + hf st nd y MID nd = y IN + h F nd y NXT nd = y IN + hf nd rd y NXT rd = y IN + hf rd 4th y NXT 4th = y IN + hf 4th The final resultant estimate of y(t + h) given by one step in Runge- Kutta then is y NXT = yst NXT + ynd NXT + yrd NXT + y4th NXT 6 With this description of the algorithm, we are ready for: Proof of Theorem A: First note that replacing (v, x) by ( v, x) leaves the differential equation unchanged Every step in Runge- Kutta also picks up a factor of, as does the conclusion of Theorem A So once we have proven the theorem for < v < bh, the case > v > bh follows immediately We start by specializing the tabular description of one step in Runge-Kutta to the ODE system in Theorem A

4 Substep Value of F st F st = ( b, v ) nd F nd = (b, v bh ) rd 4th F rd = ( b, v + bh ) F 4th = (b, v bh) Substep Estimate of y(t + h ) Estimate of y(t + h) st y MID st = (v bh, x + hv ) yst NXT = (v bh, x + hv ) nd y MID nd = (v + bh, x + h (v bh )) ynd NXT = (v + bh, x + h(v bh )) rd y NXT rd = (v bh, x + h(v + bh )) 4th y NXT 4th = (v + bh, x + h(v bh)) The proof of theorem A is now completed by inspecting the last column of the above table and observing that ( ) y st NXT + y NXT nd + y NXT rd + y NXT 4th = (v, x + h(v bh 6 6 )) What about other values of v? When v =, numerical integration by this method will leave (v, x ) unchanged; we have an equilibrium point of a discontinuous vector field For v > bh, numerical integration avoids the line v = and simply produces the expected answer (v bh, x + h(v bh )) The other cases are covered below Theorem B: Let b > and consider the system of ODE s (v, x ) = ( b sgn(v), v) with initial condition (v(t ), x(t )) = (v, x ) Then a) For any step size h > with bh < v < bh, one step of fourth order Runge Kutta changes (v, x ) to ( v sgn(v ) bh (, x + h v sgn(v ) bh )) b) For a step size h > with v = ±bh, one step of fourth order Runge Kutta changes (v, x ) to ( ( )) v 6, x v + h 4

5 c) For a step size h > with v = ± bh, one step of fourth order Runge Kutta changes (v, x ) to ( v, x + h ( v )) The proofs are readily constructed by adjusting the entries in the proof of Theorem A to these slightly different v values These answers are of a similar character to theorem A above But since v is not preserved, they only apply to one step and so will not be as striking if encountered Also, since equality of real numbers is unstable in a floating point environment, there is no reason to expect the cases v = ±bh, v = ± bh, or v = to readily show up in practice Another question one may ask is what happens if the second order system is generalized to the d dimensional system in y = (y, y,, y d ): y = b sgn(y ) y = y = y d = y d Here y plays the role of the earlier v, and y = x And so y IN replaces what we earlier denoted by v Theorem D: Let b > and y IN an initial velocity satisfying < y IN < bh Then one step of fourth order Runge Kutta changes y from y IN = (y IN, y IN,, y INd ) 5

6 to y NXT = y IN + h(, y IN, y IN,, y INd ) + h (, b 6, 5 6 y IN, 5 6 y IN,, 5 6 y INd ) + h (,, b 6, y IN, y IN,, y INd ) + h 4 (,,, b 6, 6 y IN, 6 y IN,, 6 y INd 4) With the notation we used earlier, the above is easy to establish once one expands the terms in the proof of Theorem A in powers of h Recurrences are immediate which determine the h p coefficients of Y NXT r from the h p coefficients of Y NXT r The pictures in this note were made with the older software program dstool [] In fact the author first noted the phenomenon described here in the course of analyzing some bug reports (of a more complicated example) from that system References [] A Back, J Guckenheimer, M Myers, FJ Wicklin, P Worfolk, dstool: Computer Assisted Exploration of Dynamical Systems, Notices of the American Mathematical Society 9 (99) 9 [] J Stoer, R Bulirsch, Introduction to Numerical Analysis, Springer- Verlag, New York, 98 6