We suggest a paradigm shift for the numerical integration of dynamical flows in nonlinear dynamics. Rather than use algorithms based on stepping the independent variable, typically time, we investigate the consequences of stepping in any future spacetime direction. In particular, stepping the dependent variable, typically space, can be remarkably successful, and that spatial success undermines the priority of time.
Nonlinear dynamical flows are often numerically integrated by algorithms that step the temporal independent variable, perhaps with a variable or adaptive step to control the corresponding change in the spatial dependent variable. If the goal of these algorithms is to regularize the step in the dependent variable, why not simply step the dependent variable directly (and infer the step in the independent variable indirectly)?
We systematically study the consequences of first-order stepping in arbitrary future spacetime directions for one-dimensional flows. We identify three important special cases, estimate their local errors, and compare their performances on an important initial value problem. We show that dependent variable stepping can be spectacularly successful and, by deemphasizing time, can entail a paradigm shift in integration schemes.
Important special cases include the following: Independent Variable Stepping (IVS) takes constant steps in time; Arc Length Stepping (ALS) takes constant steps tangent to the flow; Dependent Variable Stepping (DVS) takes constant steps in space.
In a spacetime diagram, the popular IVS concentrates steps on slow, shallow segments, the rival DVS (which can be pronounced “devious”) concentrates steps on fast, steep segments, and intermediate ALS steps uniformly along the flow to concentrate in spacetime at corners and other regions of high curvature.
Watch your step: numerical integration of nonlinear dynamical flows by independent variable stepping is not your only option and may not be your best option. That depends on the initial value problem and the spacetime shape of its solution.
DVS is conceptually interesting because it can be rewritten to deemphasize time. For example, in charging a capacitor, charge change becomes primary, time change becomes secondary. In deemphasizing time, DVS better describes physics.
Apart from practical benefits of using DVS for situations where it can concentrate its steps at times of changing velocity or acceleration, there is an important conceptual benefit: alternative integration schemes like DVS (and ALS) unshackle us from the hegemony of time.
In a spacetime plot, IVS integration steps bunch up at steep slopes, DVS steps bunch up at shallow slopes, and ALS steps are uniformly distributed along the flow.
When integrating the nonuniform oscillator, DVS concentrates its steps at jumps, which start and stop with high acceleration, and thereby handles them better than IVS, which squanders its steps on plateaus.
In this spacetime plot, DVS follows a particular differential flow to the same accuracy as IVS using far fewer but “intelligently placed” integration steps.
Density of IVS integration steps is constant in time, but density of DVS integration steps is constant in space and concentrated in time when needed most for important dynamical systems like the nonuniform oscillator.