Simplify Runge Kutta solver

Author: emstoudenmire

src/solvers/local_solvers/runge_kutta.jl

@@ -1,25 +1,12 @@
 
-function runge_kutta_2(H, t, ψ0)
-  Hψ = H(ψ0)
-  H2ψ = H(Hψ)
-  return (ψ0 + t * Hψ + (t^2 / 2) * H2ψ)
-end
-
-function runge_kutta_4(H, t, ψ0)
-  k1 = H(ψ0)
-  k2 = k1 + (t / 2) * H(k1)
-  k3 = k1 + (t / 2) * H(k2)
-  k4 = k1 + t * H(k3)
-  return ψ0 + (t / 6) * (k1 + 2 * k2 + 2 * k3 + k4)
-end
-
-function runge_kutta_solver(H, time, ψ; order=4, kws...)
-  if order == 4
-    Hψ = runge_kutta_4(H, time, ψ)
-  elseif order == 2
-    Hψ = runge_kutta_2(H, time, ψ)
-  else
-    error("For runge_kutta_solver, must specify `order` keyword")
+function runge_kutta_solver(H, t, ψ; order=4, kws...)
+  # For linear ODE, Runge-Kutta is a Taylor series.
+  # Pattern below derived as:
+  # exp(tH)ψ = ψ + tHψ +  (tH)^2(ψ)/2! + (tH)^3(ψ)/3! + ...
+  # = ψ + (tH) * (ψ + (tH)/2 * (ψ + (tH)/3 * (ψ + ...)))
+  eHψ = copy(ψ)
+  for ord in reverse(1:order)
+    eHψ = (t/ord)*H(eHψ) + ψ
   end
-  return Hψ, (;)
+  return eHψ, (;)
 end