I'm using a simple version of Verlet integration for a particleparticle interaction system with collisions. At the end of each iteration, I integrate like this:verletX = ( 1.92*x[i] - 0.92*x_old[i] ) + fx[i]*0.002;verletY = ( 1.92*y[i] - 0.92*y_old[i] ) + fy[i]*0.002;x_old[i] = x[i];y_old[i] = y[i];x[i] = verletX;y[i] = verletY;Can I increase its order of error (to have more precision for decreasing timesteps) using even more old data of X and Y? For now it should be less precise than a RK4, but Im sure its much better than a simple Euler integration. For example can I use the data from the last eight iterations to reach RK4 levels?The reason Im not trying RK4 is that the calculation of forces takes too much time. However, having a copy of the old data and updating it does not. Also I dont need exactness; I just need the system stability such that particles stay in fixed points rather than having explosive close range leaps.
Higher-order Verlet integration
computational physics;time integration;runge kutta;particle
Use multistep methods in this case. See the Adams-Bashforth method if it's nonstiff, or Adams-Bashforth-Moulton methods, or if its stiff BDF methods. These use past timepoints like you want in order to increase the order of accuracy.I would highly recommend checking out something like Sundials which has variable timestep plus variable order methods. This helps a lot with the stability of these methods and lets them get a lot more efficiency out.