Stability of stabilized rk2 (Longcope's trick, rk2plus.m) for the scalar wave equation.
x = dimensionless timestep, hc/Delta
y = epsilon, which is used to modify the half step to (1 + epsilon) h/2.

The stable region is in gray:

y>[1+y]^2*x^2

A conservative rule of thumb for choosing epsilon. It is easy to show that rk2plus is still second order for epsilon below this curve:

y=2*x^2,abs(x)<0.4

A more careful fit to the stability region, giving even lower epsilon values for a given dimensionless timestep:

y=x^2+4*x^4,abs(x)<0.5

Particular values used for the movies in the parent directory:

d=0.2

vector(x,y)=vector(d,d^2+4*d^4)

b=0.4

vector(x,y)=vector(b,b^2+4*b^4)

Graph of the formula

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