def elastic_RK4(rv, rs, v, s, rho, mu, nx, dx, order, y, t, dt, r0, r1, tau0_1, tau0_2, tauN_1, tauN_2, type_0, forcing):
# fourth order Runge-Kutta time-stepping
import rate
import numpy as np
# intialize arrays for Runge-Kutta stages
k1v = np.zeros((nx, 1))
k1s = np.zeros((nx, 1))
k2v = np.zeros((nx, 1))
k2s = np.zeros((nx, 1))
k3v = np.zeros((nx, 1))
k3s = np.zeros((nx, 1))
k4v = np.zeros((nx, 1))
k4s = np.zeros((nx, 1))
rate.elastic_rate(k1v, k1s, v, s, rho, mu, nx, dx, order, t, y, r0, r1, tau0_1, tau0_2, tauN_1, tauN_2, type_0, forcing)
rate.elastic_rate(k2v, k2s, v+0.5*dt*k1v, s+0.5*dt*k1s, rho, mu, nx, dx, order, t+0.5*dt, y, r0, r1, tau0_1, tau0_2, tauN_1, tauN_2, type_0, forcing)
rate.elastic_rate(k3v, k3s, v+0.5*dt*k2v, s+0.5*dt*k2s, rho, mu, nx, dx, order, t+0.5*dt, y, r0, r1, tau0_1, tau0_2, tauN_1, tauN_2, type_0, forcing)
rate.elastic_rate(k4v, k4s, v+dt*k3v, s+dt*k3s, rho, mu, nx, dx, order, t+dt, y, r0, r1, tau0_1, tau0_2, tauN_1, tauN_2, type_0, forcing)
# update fields
rv[:,:] = v + (dt/6.0)*(k1v + 2.0*k2v + 2.0*k3v + k4v)
rs[:,:] = s + (dt/6.0)*(k1s + 2.0*k2s + 2.0*k3s + k4s)
def elastic_RK4(rv_l, rs_l, v_l, s_l, rho_l, mu_l, nx_l, dx_l, order_l, y_l, t_l, dt_l, r0_l, r1_l, tau0_1_l, tau0_2_l, tauN_1_l, tauN_2_l, rv_r, rs_r, v_r, s_r, rho_r, mu_r, nx_r, dx_r, order_r, y_r, t_r, dt_r, r0_r, r1_r, tau0_1_r, tau0_2_r, tauN_1_r, tauN_2_r, rslip, rpsi, slip, psi,friction_parameters):
nx_r = nx_l
dx_r = dx_l
order_r = order_l
t_r = t_l
dt_r = dt_l
# fourth order Runge-Kutta time-stepping
import rate
import numpy as np
# intialize arrays for Runge-Kutta stages
k1v_l = np.zeros((nx_l, 1))
k1s_l = np.zeros((nx_l, 1))
k2v_l = np.zeros((nx_l, 1))
k2s_l = np.zeros((nx_l, 1))
k3v_l = np.zeros((nx_l, 1))
k3s_l = np.zeros((nx_l, 1))
k4v_l = np.zeros((nx_l, 1))
k4s_l = np.zeros((nx_l, 1))
k1v_r = np.zeros((nx_r, 1))
k1s_r = np.zeros((nx_r, 1))
k2v_r = np.zeros((nx_r, 1))
k2s_r = np.zeros((nx_r, 1))
k3v_r = np.zeros((nx_r, 1))
k3s_r = np.zeros((nx_r, 1))
k4v_r = np.zeros((nx_r, 1))
k4s_r = np.zeros((nx_r, 1))
k1slip = np.zeros((1, 1))
k1psi = np.zeros((1, 1))
k2slip = np.zeros((1, 1))
k2psi = np.zeros((1, 1))
k3slip = np.zeros((1, 1))
k3psi = np.zeros((1, 1))
k4slip = np.zeros((1, 1))
k4psi = np.zeros((1, 1))
rate.elastic_rate(k1v_l, k1s_l, v_l, s_l, rho_l, mu_l, nx_l, dx_l, order_l, t_l, y_l, r0_l, r1_l, tau0_1_l, tau0_2_l, tauN_1_l, tauN_2_l, k1v_r, k1s_r, v_r, s_r, rho_r, mu_r, nx_r, dx_r, order_r, t_r, y_r, r0_r, r1_r, tau0_1_r, tau0_2_r, tauN_1_r, tauN_2_r, slip, psi, k1slip, k1psi,friction_parameters)
rate.elastic_rate(k2v_l, k2s_l, v_l+0.5*dt_l*k1v_l, s_l+0.5*dt_l*k1s_l, rho_l, mu_l, nx_l, dx_l, order_l, dt_l+0.5*dt_l, y_l, r0_l, r1_l, tau0_1_l, tau0_2_l, tauN_1_l, tauN_2_l, k2v_r, k2s_r, v_r+0.5*dt_r*k1v_r, s_r+0.5*dt_r*k1s_r, rho_r, mu_r, nx_r, dx_r, order_r, dt_r+0.5*dt_r, y_r, r0_r, r1_r, tau0_1_r, tau0_2_r, tauN_1_r, tauN_2_r,
slip+0.5*dt_l*k1slip, psi+0.5*dt_l*k1psi, k2slip, k2psi,friction_parameters)
rate.elastic_rate(k3v_l, k3s_l, v_l+0.5*dt_l*k2v_l, s_l+0.5*dt_l*k2s_l, rho_l, mu_l, nx_l, dx_l, order_l, dt_l+0.5*dt_l, y_l, r0_l, r1_l, tau0_1_l, tau0_2_l, tauN_1_l, tauN_2_l, k3v_r, k3s_r, v_r+0.5*dt_r*k2v_r, s_r+0.5*dt_r*k2s_r, rho_r, mu_r, nx_r, dx_r, order_r, dt_r+0.5*dt_r, y_r, r0_r, r1_r, tau0_1_r, tau0_2_r, tauN_1_r, tauN_2_r,
slip+0.5*dt_l*k2slip, psi+0.5*dt_l*k2psi, k3slip, k3psi,friction_parameters)
rate.elastic_rate(k4v_l, k4s_l, v_l+dt_l*k3v_l, s_l+dt_l*k3s_l, rho_l, mu_l, nx_l, dx_l, order_l, dt_l+dt_l, y_l, r0_l, r1_l, tau0_1_l, tau0_2_l, tauN_1_l, tauN_2_l, k4v_r, k4s_r, v_r+dt_r*k3v_r, s_r+dt_r*k3s_r, rho_r, mu_r, nx_r, dx_r, order_r, dt_r+dt_r, y_r, r0_r, r1_r, tau0_1_r, tau0_2_r, tauN_1_r, tauN_2_r,
slip+ dt_l*k3slip, psi+dt_l*k3psi, k4slip, k4psi,friction_parameters)
# update fields
rv_l[:,:] = v_l + (dt_l/6.0)*(k1v_l + 2.0*k2v_l + 2.0*k3v_l + k4v_l)
rs_l[:,:] = s_l + (dt_l/6.0)*(k1s_l + 2.0*k2s_l + 2.0*k3s_l + k4s_l)
# update fields
rv_r[:,:] = v_r + (dt_r/6.0)*(k1v_r + 2.0*k2v_r + 2.0*k3v_r + k4v_r)
rs_r[:,:] = s_r + (dt_r/6.0)*(k1s_r + 2.0*k2s_r + 2.0*k3s_r + k4s_r)
# update slip and state
rslip[:,:] = slip + (dt_r/6.0)*(k1slip + 2.0*k2slip + 2.0*k3slip + k4slip)
rpsi[:,:] = psi + (dt_r/6.0)*(k1psi + 2.0*k2psi + 2.0*k3psi + k4psi)