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My colleague and I are trying to study the three-body problem, with different integration schemes, starting from the two-body problem. We implemented the symplectic Euler scheme and the Runge–Kutta 4th order in C++, and the trajectories obtained are the following.

Enter image description here

Enter image description here

As we can see, they are almost elliptical with Euler integration, while they become bigger and bigger with rk4, increasing the energy of the system at each revolution:

Enter image description here

Is it possible that the rk4 increases its energy or do we have something wrong in our code?

#include <iostream>
#include <cmath>
#include <array>
#include <fstream>
#include "integrators.h"
#include <string>
#include <map>

// Value-Defintions of the different String values
enum StringValue { evNotDefined,
                   evStringValue1,
                   evStringValue2
                 };

// Map to associate the strings with the enum values
static std::map<std::string, StringValue> s_mapStringValues;

void Initialize(){
  s_mapStringValues["euler"] = evStringValue1;
  s_mapStringValues["rk4"] = evStringValue2;
}

static constexpr int DIM = 4;
static constexpr double G = 10;
static constexpr int N_BODIES = 3;
static constexpr int N_STEPS = 300000;

double distance(std::array<double, 3> r1, std::array<double, 3> r2){
    return sqrt(pow(r1[0]-r2[0],2)+pow(r1[1]-r2[1],2)+pow(r1[2]-r2[2],2));
}

class Planet{
public:

    double m;
    std::array <double, 3> x;
    std::array <double, 3> v;
    std::array <double, 3> a;
    double energy;

    Planet (double mass, double x_position, double y_position, double z_position, double x_velocity, double y_velocity, double z_velocity) {
        m = mass;
            x[0] = x_position;
            x[1] = y_position;
            x[2] = z_position;
            v[0] = x_velocity;
            v[1] = y_velocity;
            v[2] = z_velocity;
        };
        void computeKineticEnergy(){
            energy = 0.5 * m * (pow(v[0],2)+pow(v[1],2)+pow(v[2],2));
        }
};


double computePotentialEnergy(Planet planet1, Planet planet2, Planet planet3){
    return -1 * G * ( planet3.m * planet1.m / distance(planet3.x, planet1.x) +  planet3.m * planet2.m / distance(planet3.x, planet2.x) + planet1.m * planet2.m / distance(planet1.x, planet2.x));

}

double acceleration(Planet A, Planet B, Planet C, int axe){
    // Compute the acceleration of the body C, specifying the axis.
    double mass_A = A.m;
    double mass_B = B.m;
    double posx_A = A.x[0];
    double posx_B = B.x[0];
    double posx_C = C.x[0];
    double posy_A = A.x[1];
    double posy_B = B.x[1];
    double posy_C = C.x[1];
    double posz_A = A.x[2];
    double posz_B = B.x[2];
    double posz_C = C.x[2];
    if (axe == 0){
        return (-1 * G * (mass_A * (posx_C-posx_A) / pow(sqrt(pow(posx_C-posx_A,2)+pow(posy_C-posy_A,2)+pow(posz_C-posz_A,2)), 3) + mass_B * (posx_C-posx_B) / pow(sqrt(pow(posx_C-posx_B,2)+pow(posy_C-posy_B,2)+pow(posz_C-posz_B,2)), 3)));
    }else if (axe == 1){
        return (-1 * G * (mass_A * (posy_C-posy_A) / pow(sqrt(pow(posx_C-posx_A,2)+pow(posy_C-posy_A,2)+pow(posz_C-posz_A,2)), 3) + mass_B * (posy_C-posy_B) / pow(sqrt(pow(posx_C-posx_B,2)+pow(posy_C-posy_B,2)+pow(posz_C-posz_B,2)), 3)));
    }else if (axe == 2){
        return (-1 * G * (mass_A * (posz_C-posz_A) / pow(sqrt(pow(posx_C-posx_A,2)+pow(posy_C-posy_A,2)+pow(posz_C-posz_A,2)), 3) + mass_B * (posz_C-posz_B) / pow(sqrt(pow(posx_C-posx_B,2)+pow(posy_C-posy_B,2)+pow(posz_C-posz_B,2)), 3)));
    }
}

double F(double x, double v, double t, Planet A, Planet B, Planet C, int j ){
    // Function to integrate via Runge-Kutta.
    double mass_A = A.m;
    double mass_B = B.m;
    double posx_A = A.x[0];
    double posx_B = B.x[0];
    double posx_C = C.x[0];
    double posy_A = A.x[1];
    double posy_B = B.x[1];
    double posy_C = C.x[1];
    double posz_A = A.x[2];
    double posz_B = B.x[2];
    double posz_C = C.x[2];

    if (j == 0){
        return (-1 * G * (mass_A * (x-posx_A) / pow(sqrt(pow(x-posx_A,2)+pow(posy_C-posy_A,2)+pow(posz_C-posz_A,2)), 3) + mass_B * (x-posx_B) / pow(sqrt(pow(x-posx_B,2)+pow(posy_C-posy_B,2)+pow(posz_C-posz_B,2)), 3)));
    }else if (j == 1) {
        return (-1 * G * (mass_A * (x-posy_A) / pow(sqrt(pow(posx_C-posx_A,2)+pow(x-posy_A,2)+pow(posz_C-posz_A,2)), 3) + mass_B * (x-posy_B) / pow(sqrt(pow(posx_C-posx_B,2)+pow(x-posy_B,2)+pow(posz_C-posz_B,2)), 3)));
    }else if (j == 2) {
        return (-1 * G * (mass_A * (x-posz_A) / pow(sqrt(pow(posx_C-posx_A,2)+pow(posy_C-posy_A,2)+pow(x-posz_A,2)), 3) + mass_B * (x-posz_B) / pow(sqrt(pow(posx_C-posx_B,2)+pow(posy_C-posy_B,2)+pow(x-posz_B,2)), 3)));
    }
}

int main(int argc, char** argv){

    double h = 0.0005;

    Planet A(100, -10, 10, -11, -3, 0, 0);
    Planet B(100, 0, 0, 0, 3, 0, 0);
    Planet C(0, 10, 14, 12, 3, 0, 0);

    double x_A[DIM][3];
    double x_B[DIM][3];
    double x_C[DIM][3];
    double vx_A;
    double vy_A;
    double vz_A;
    double vx_B;
    double vy_B;
    double vz_B;
    double vx_C;
    double vy_C;
    double vz_C;

    double mass_A = A.m;
    double mass_B = B.m;
    double mass_C = C.m;

    x_A[0][0] = A.x[0];
    x_B[0][0] = B.x[0];
    x_C[0][0] = C.x[0];

    vx_A = A.v[0];
    vx_B = B.v[0];
    vx_C = C.v[0];

    x_A[1][0] = A.x[1];
    x_B[1][0] = B.x[1];
    x_C[1][0] = C.x[1];

    vy_A = A.v[1];
    vy_B = B.v[1];
    vy_C = C.v[1];

    x_A[2][0] = A.x[2];
    x_B[2][0] = B.x[2];
    x_C[2][0] = C.x[2];

    vz_A = A.v[2];
    vz_B = B.v[2];
    vz_C = C.v[2];

    Initialize();

    std::ofstream file_energy("Total_energy_" + std::string(argv[1]) + ".csv");
    std::ofstream output_file_A("positions_A_" + std::string(argv[1]) + ".csv");
    std::ofstream output_file_B("positions_B_" + std::string(argv[1]) + ".csv");
    std::ofstream output_file_C("positions_C_" + std::string(argv[1]) + ".csv");


    output_file_A<<"x;y;z"<<std::endl;
    output_file_B<<"x;y;z"<<std::endl;
    output_file_C<<"x;y;z"<<std::endl;
    file_energy<<"k;p"<<std::endl;

    double m1;
    double k1;
    double m2;
    double k2;
    double m3;
    double k3;
    double m4;
    double k4;

    std::array<double,3> vA = A.v; //condizione iniziale velocita
    std::array<double,3> xA = A.x;
    std::array<double,3> vB = B.v; //condizione iniziale velocita
    std::array<double,3> xB = B.x;
    std::array<double,3> vC = C.v; //condizione iniziale velocita
    std::array<double,3> xC = C.x;
    double t;
    std::array<double,3> cm = computeCM(A, B, C);

    if (argc>=2){
        switch (s_mapStringValues[argv[1]]){
            case evStringValue1:
                // ==========================================================
                //                          EULER
                // ==========================================================

                for (int i=0; i<N_STEPS-1; i++){
                    output_file_A << A.x[0] << ";" << A.x[1] << ";" << A.x[2]<< std::endl;
                    output_file_B << B.x[0] << ";" << B.x[1] << ";" << B.x[2]<< std::endl;
                    output_file_C << C.x[0] << ";" << C.x[1] << ";" << C.x[2]<< std::endl;
                    for(int j=0; j<DIM-1; j++){

                        A.a[j] = acceleration(B, C, A, j);
                        B.a[j] = acceleration(A, C, B, j);
                        C.a[j] = acceleration(B, A, C, j);

                        A.v[j] += A.a[j] * h;
                        B.v[j] += B.a[j] * h;
                        C.v[j] += C.a[j] * h;

                        x_A[j][0] = x_A[j][0] + A.v[j] * h;
                        x_B[j][0] = x_B[j][0] + B.v[j] * h;
                        x_C[j][0] = x_C[j][0] + C.v[j] * h;

                    }
                    for (int j=0;j<DIM-1;j++){
                        A.x[j] = x_A[j][0];
                        B.x[j] = x_B[j][0];
                        C.x[j] = x_C[j][0];
                    }

                    A.computeKineticEnergy();
                    B.computeKineticEnergy();
                    C.computeKineticEnergy();
                    cm = computeCM(A,B,C);
                    file_energy<<A.energy + B.energy + C.energy<<";"<< computePotentialEnergy(A, B, C)<<std::endl;

                }

                break;
            case evStringValue2:{
                // ==========================================================
                //                          RUNGE KUTTA 4
                // ==========================================================

                for(int i=0; i<N_STEPS-1; i++){
                    for(int j=0; j<DIM-1; j++){
                        // body A
                        m1 = h*vA[j];
                        k1 = h*F(xA[j], vA[j], t, C, B, A, j);

                        m2 = h*(vA[j] + 0.5*k1);
                        k2 = h*F(xA[j]+0.5*m1, vA[j]+0.5*k1, t+0.5*h, C, B, A, j);

                        m3 = h*(vA[j] + 0.5*k2);
                        k3 = h*F(xA[j]+0.5*m2, vA[j]+0.5*k2, t+0.5*h, C, B, A, j);

                        m4 = h*(vA[j] + k3);
                        k4 = h*F(xA[j]+m3, vA[j]+k3, t+h, C, B, A, j);

                        xA[j] += (m1 + 2*m2 + 2*m3 + m4)/6;
                        vA[j] += (k1 + 2*k2 + 2*k3 + k4)/6;
                    }

                    for(int j=0; j<DIM-1; j++){
                        // body B
                        m1 = h*vB[j];
                        k1 = h*F(xB[j], vB[j], t, A, C, B, j);  //(x, v, t)

                        m2 = h*(vB[j] + 0.5*k1);
                        k2 = h*F(xB[j]+0.5*m1, vB[j]+0.5*k1, t+0.5*h, A, C, B, j);

                        m3 = h*(vB[j] + 0.5*k2);
                        k3 = h*F(xB[j]+0.5*m2, vB[j]+0.5*k2, t+0.5*h, A, C, B, j);

                        m4 = h*(vB[j] + k3);
                        k4 = h*F(xB[j]+m3, vB[j]+k3, t+h, A, C, B, j);

                        xB[j] += (m1 + 2*m2 + 2*m3 + m4)/6;
                        vB[j] += (k1 + 2*k2 + 2*k3 + k4)/6;
                    }

                    for(int j=0; j<DIM-1; j++){
                        // body C
                        m1 = h*vC[j];
                        k1 = h*F(xC[j], vC[j], t, A, B, C, j);  //(x, v, t)

                        m2 = h*(vC[j] + 0.5*k1);
                        k2 = h*F(xC[j]+0.5*m1, vC[j]+0.5*k1, t+0.5*h, A, B, C, j);

                        m3 = h*(vC[j] + 0.5*k2);
                        k3 = h*F(xC[j]+0.5*m2, vC[j]+0.5*k2, t+0.5*h, A, B, C, j);

                        m4 = h*(vC[j] + k3);
                        k4 = h*F(xC[j]+m3, vC[j]+k3, t+h, A, B, C, j);

                        xC[j] += (m1 + 2*m2 + 2*m3 + m4)/6;
                        vC[j] += (k1 + 2*k2 + 2*k3 + k4)/6;
                    }

                    for(int j=0; j<DIM-1;j++){
                        A.v[j] = vA[j];
                        B.v[j] = vB[j];
                        C.v[j] = vC[j];
                        A.x[j] = xA[j];
                        B.x[j] = xB[j];
                        C.x[j] = xC[j];
                    }
                    output_file_A << A.x[0] << ";" << A.x[1] << ";" << A.x[2]<< std::endl;
                    output_file_B << B.x[0] << ";" << B.x[1] << ";" << B.x[2]<< std::endl;
                    output_file_C << C.x[0] << ";" << C.x[1] << ";" << C.x[2]<< std::endl;
                    A.computeKineticEnergy();
                    B.computeKineticEnergy();
                    C.computeKineticEnergy();
                    file_energy<<A.energy + B.energy + C.energy <<";"<< computePotentialEnergy(A, B, C)<<std::endl;
                }
                }
                break;

            default:
                std::cout<<"Insert an argument between: euler, or rk4"<<std::endl;
                return 0;
        }
    }else{
        std::cout<<"Insert an argument between: euler, or rk4"<<std::endl;
        return 0;
    }

//----------------------------------------------------------------

    output_file_A.close();
    output_file_B.close();
    output_file_C.close();
    file_energy.close();

    return 0;
}

edit:

I think I have understood the problem. I have modified the code as follow, but now the energy decreases a lot and the trajectories become more and more little.

for(int i=0; i<N_STEPS-1; i++){
                    for(int j=0; j<DIM-1; j++){
                        // body A
                        m1[0][j] = h*vA[j];
                        k1[0][j] = h*F(xA[j], vA[j], t, C, B, A, j);
                        // body B
                        m1[1][j] = h*vA[j];
                        k1[1][j] = h*F(xA[j], vA[j], t, C, A, B, j); 
                        // body C
                        m1[2][j] = h*vA[j];
                        k1[2][j] = h*F(xA[j], vA[j], t, A, B, C, j); 

                    }
                    for(int j=0; j<DIM-1;j++){
                        A.v[j] += 0.5 * k1[0][j];
                        B.v[j] += 0.5 * k1[1][j];
                        C.v[j] += 0.5 * k1[2][j];          
                        A.x[j] += 0.5 * m1[0][j];
                        B.x[j] += 0.5 * m1[1][j];
                        C.x[j] += 0.5 * m1[2][j];
                    }
                    for(int j=0; j<DIM-1; j++){
                        //Body A
                        m2[0][j] = h*(A.v[j]);
                        k2[0][j] = h*F(A.x[j], A.v[j], t+0.5*h, C, B, A, j);
                        //Body B
                        m2[1][j] = h*(B.v[j]);
                        k2[1][j] = h*F(B.x[j], B.v[j], t+0.5*h, C, A, B, j);
                        // Body C
                        m2[2][j] = h*(C.v[j]);
                        k2[2][j] = h*F(C.x[j], C.v[j], t+0.5*h, A, B, C, j);

                    }
                     for(int j=0; j<DIM-1;j++){
                        A.v[j] += 0.5 * k2[0][j];
                        B.v[j] += 0.5 * k2[1][j];
                        C.v[j] += 0.5 * k2[2][j];          
                        A.x[j] += 0.5 * m2[0][j];
                        B.x[j] += 0.5 * m2[1][j];
                        C.x[j] += 0.5 * m2[2][j];
                    }
                    for(int j=0; j<DIM-1; j++){
                        //Body A
                        m3[0][j] = h*(A.v[j]);
                        k3[0][j] = h*F(A.x[j], A.v[j], t+0.5*h, C, B, A, j);
                       
                        //Body B
                        m3[1][j] = h*(B.v[j]);
                        k3[1][j] = h*F(B.x[j], B.v[j], t+0.5*h, C, A, B, j);
                        // Body C
                        m3[2][j] = h*(C.v[j]);
                        k3[2][j] = h*F(C.x[j], C.v[j], t+0.5*h, A, B, C, j);
                    }
                     for(int j=0; j<DIM-1;j++){
                        A.v[j] += k3[0][j];
                        B.v[j] += k3[1][j];
                        C.v[j] += k3[2][j];          
                        A.x[j] += m3[0][j];
                        B.x[j] += m3[1][j];
                        C.x[j] += m3[2][j];
                    }
                    for(int j=0; j<DIM-1; j++){
                        //Body A
                        m4[0][j] = h*(A.v[j]);
                        k4[0][j] = h*F(A.x[j], A.v[j], t + h, C, B, A, j);
                        //Body B
                        m4[1][j] = h*(B.v[j]);
                        k4[1][j] = h*F(B.x[j], B.v[j], t + h, C, A, B, j);
                        // Body C
                        m4[2][j] = h*(C.v[j]);
                        k4[2][j] = h*F(C.x[j], C.v[j], t + h, A, B, C, j);
                    }
                    for(int j=0; j<DIM-1; j++){
                        xA[j] += (m1[0][j] + 2*m2[0][j] + 2*m3[0][j] + m4[0][j])/6;
                        vA[j] += (k1[0][j] + 2*k2[0][j] + 2*k3[0][j] + k4[0][j])/6;
                        xB[j] += (m1[1][j] + 2*m2[1][j] + 2*m3[1][j] + m4[1][j])/6;
                        vB[j] += (k1[1][j] + 2*k2[1][j] + 2*k3[1][j] + k4[1][j])/6;
                        xC[j] += (m1[2][j] + 2*m2[2][j] + 2*m3[2][j] + m4[2][j])/6;
                        vC[j] += (k1[2][j] + 2*k2[2][j] + 2*k3[2][j] + k4[2][j])/6;
                    }



   for(int j=0; j<DIM-1;j++){
                        A.v[j] = vA[j];
                        B.v[j] = vB[j];
                        C.v[j] = vC[j];          
                        A.x[j] = xA[j];
                        B.x[j] = xB[j];
                        C.x[j] = xC[j];
                    }

Edit 2;

Energy of the system with the proposed solution: [enter image description here]

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  • $\begingroup$ one thing to note is that rk4 is a pretty bad solver. just changing the constants can give you much lower errors, and there are dinners of easy higher order that do very well for orbits $\endgroup$ Nov 21, 2022 at 3:50
  • $\begingroup$ As a general rule using a sympletic method, which RK4 isn't, is best when there is little to no "friction" in your system. Friction can be resistors, fluid viscosity, dry friction, etc. 4th order symplectic methods exist. $\endgroup$ Nov 21, 2022 at 20:35
  • $\begingroup$ @KevinKostlan : If there is friction the system is not conservative, so there is no energy etc. to conserve. Then Verlet is just another 2nd order method, the 4th order modification likewise just another 4th order method. $\endgroup$ Nov 21, 2022 at 22:18
  • $\begingroup$ @LutzLehmann: Yes that is important to keep in mind. I consider a method "sympletic" if it becomes time-reversable in the limit of zero friction. Verlet is an example but not RK4. $\endgroup$ Nov 21, 2022 at 23:23
  • $\begingroup$ It's a vague memory, but I recall a method based on invariants. In this case total energy. You basically construct an operator that generates the next time step by stepping each particle in a way that conserves all the required conserved quantities. For total energy it is the Hamiltonian. Sorry to be so vague. $\endgroup$
    – Boba Fit
    Nov 21, 2022 at 23:26

2 Answers 2

13
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RK4 is not symplectic so it has no guarantee of energy conservation. Especially when solving an N-body problem where two bodies pass by close to each other the energy conservation can be violated quite drastically. That being said, you can mitigate the energy conservation problem by reducing $\Delta t$. At some point you won't get these massive steps in total energy, but instead a small-ish steady trend up or down.

The typical way to check if you've implemented a numerical method correctly is to check its order of accuracy by comparing the solution at various resolutions. For example, if you halve $\Delta t$ RK4 should improve the solution by $\Delta t^4$ compared to the analytical solution.

edit:

As pointed out by Lutz Lehmann, you implemented the RK4 method incorrectly. Specifically, these lines of code:

for (int j = 0; j < DIM - 1; j++)
{
  // body A
  m1 = h * vA[j];
  k1 = h * F(xA[j], vA[j], t, C, B, A, j);

  m2 = h * (vA[j] + 0.5 * k1);
  k2 = h * F(xA[j] + 0.5 * m1, vA[j] + 0.5 * k1, t + 0.5 * h, C, B, A, j);

  m3 = h * (vA[j] + 0.5 * k2);
  k3 = h * F(xA[j] + 0.5 * m2, vA[j] + 0.5 * k2, t + 0.5 * h, C, B, A, j);

  m4 = h * (vA[j] + k3);
  k4 = h * F(xA[j] + m3, vA[j] + k3, t + h, C, B, A, j);

  xA[j] += (m1 + 2 * m2 + 2 * m3 + m4) / 6;
  vA[j] += (k1 + 2 * k2 + 2 * k3 + k4) / 6;
}

// ... same for body B and C

for (int j = 0; j < DIM - 1; j++)
{
  A.v[j] = vA[j];
  B.v[j] = vB[j];
  C.v[j] = vC[j];
  A.x[j] = xA[j];
  B.x[j] = xB[j];
  C.x[j] = xC[j];
}

We'll focus specifically on what you're doing to compute body A. Notice that when you go to compute k2, you're using the original position of body B and C because you haven't computed their new position. The correct formulation for RK4 in coupled systems is (just one RK4 method): $$ \vec{K}_1 = \vec{F}(\vec{q}_0)\\ \vec{K}_2 = \vec{F}(\vec{q}_0 + \Delta t \frac{\vec{K}_1}{2})\\ \vec{K}_3 = \vec{F}(\vec{q}_0 + \Delta t \frac{\vec{K}_2}{2})\\ \vec{K}_4 = \vec{F}(\vec{q}_0 + \Delta t \vec{K}_4) $$ The size of each of these vectors for 3D N-body is $3 N$, not just $3$. So while you are computing the 3 components of $K_1$ associated with body A, you are still missing these for body B and C, so when you try to compute $K_2$ for body A this calculation is incorrect. A similar problem occurs for the other bodies.

There are similar problems in your function which calculates the forces, since it's referencing positions relative to the Planet objects A, B, and C which don't get updated until the very end so you are not using any intermediate information in the latter stages.

Refer to Lutz's answer for an example of what the correct implementation looks like in Python and hopefully you can adapt that to your C++ code.

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    $\begingroup$ I would add to this that Geometric Numerical Integration is a great reference on symplectic methods if you want to read more. $\endgroup$ Nov 20, 2022 at 17:59
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    $\begingroup$ Also note that there are no symplectic explicit methods: scicomp.stackexchange.com/questions/32728/… - so if you symplecticity is important for you, you will have to switch to implicit methods. $\endgroup$
    – Daniel
    Nov 21, 2022 at 7:27
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    $\begingroup$ An incorrect implementation would also have bad energy conservation properties... though it's certainly true that RK4 does suffer from these problems (maybe not to this degree). Proper convergence testing is much better at identifying whether the problem is really the implementation or something more fundamental with the chosen discretization. $\endgroup$ Nov 21, 2022 at 8:16
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    $\begingroup$ @Daniel luckily N-body falls into the category of problems you can implement some explicit symplectic integrators. Though even without the symplectic property standard RK methods can have energy conservation on the order of 1e-12, which while not perfect or to floating point precision is usually good enough for modest time ranges. $\endgroup$ Nov 21, 2022 at 8:20
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    $\begingroup$ @helloworld922 : Symplectic methods require a fixed step size to have their advantageous properties. And that also only as long as they stay far away from the singular points of the vector field. In contrast, to get good performance from RK methods they need to be variable-step, the step sizes adapted to an error tolerance. Then the conserved quantities will also be correct to this level, possibly a little bit better. $\endgroup$ Nov 21, 2022 at 9:22
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Main error

As I pointed out in the previous questions of this series

the primary problem is that the methods are not implemented correctly. The loop order in the method step is always "outer loop - stages, inner loop - components" not "outer - components, inner - stages". In the latter wrong variants, in the coupled equations the required updates for the later components will be missing.

As commented, the actually implemented loop structure in OP is "bodies - stages - some components", which uses a mix of previously updated data and unmodified data of later bodies. This still misses that the stage 2 of body A should use the stage 1 data of bodies B and C, which however do not exist and are not requested in OP code. That is, if you have any moving, dynamical variable $u$, then in the computation of stage 2 data (similar the later stages) you need to use either the value u+0.5*k1_u with the derivative from the earlier stage, or $u(t+h/2)$ from either a fixed formula or some inter- or extrapolation of high enough order.

One could actually separate the integration of the bodies in this way, but then one would have to interpolate and extrapolate the data of the other bodies. This could even be arranged so that each body has its own adaptive step size.

If everything else is correctly implemented, these wrong methods at least stay consistent, thus of order 1. For very small step sizes one can thus still get quantitatively correct results.

Reference results

Correctly implemented and with the provided step size, there is no visible error in the RK4 results, while the Verlet energy has spikes downward for each orbit of size 0.5 from the basis level of $-4681.5$. There are over the span of $Δt=150$ some $30$ orbits.

This is clearly different from your plot where the spikes of the Verlet method have amplitude about $100$ and the "RK4" labeled method increases the energy by 2600.

To get a visible staircase for the energy of RK4, use the 4 times larger step size of $h=0.002$ with 75000 steps for Verlet, and to keep the complexity in terms of the number of function evaluations constant, $h_{RK4}=4h$ with for the RK4 method with $18750$ steps.

detail zoom
enter image description here enter image description here

For Verlet, the energy has spikes of $0.5$ up and $3.5$ down. The RK4 energy falls in the same time by a smidgen more than $1$.

Comparing some graphs, the Verlet energy has a periodic pattern with spikes $O(h^2)$, while the RK4 energy falls like $O(th^5)$.

Reference implementation in Python

There is no easy solution for a composite model, but since here all components are of the same type, one can implement the state of the system as a 2-dimensional array. In Python it is easy to transform such into flat vectors and back.

Data of the system

G = 10;

A=(100, -10, 10, -11, -3, 0, 0);   
B=(100, 0, 0, 0, 3, 0, 0);
C=(0, 10, 14, 12, 3, 0, 0);
bodies=np.array([A,B,C])

m = bodies[:,0]
x0 = bodies[:,1:4]
v0 = bodies[:,4:7]

Functions of the model

Derivatives

def acc(x):
    dx = x[None,:,:]-x[:,None,:]
    r3 = np.sum(dx**2,axis=2)**1.5
    return -G*np.sum(m[:,None,None]*dx/(1e-40+r3[:,:,None]), axis=0)
    

def sys(t,u):
    x,v = u.reshape([2,-1,3])
    return np.reshape([v,acc(x)],-1)

Energy

def kineticEnergy(m,v):
    imp = 0.5*m * np.sum(v**2,axis=1).T
    return np.sum(imp.T, axis=0)

def potentialEnergy(m1,x1,m2,x2):
    dx = x2-x1
    r = np.sum(dx**2,axis=0)**0.5
    return -G*m1*m2/r

def totalEnergy(m,x,v):
    pot = 0
    for i in range(len(m)):
        for j in range(i):
            pot += potentialEnergy(m[i],x[i],m[j],x[j])
    return pot+kineticEnergy(m,v)

Verlet method (leapfrog implementation)

Data structures

T=150
t,h = np.linspace(0,T,500*T+1,retstep=True)

x_ver = np.zeros([len(m),3,len(t)])
v_ver = np.zeros([len(m),3,len(t)+1])
x_ver[:,:,0] = xi = x0
a0 = acc(x0)
v_ver[:,:,0] =      v0-0.5*h*a0
v_ver[:,:,1] = vi = v0+0.5*h*a0

Time loop

for i in range(1,len(t)):
    x_ver[:,:,i] = xi = xi + h*vi ## position at i-1 to i using velocity at i-1/2
    a = acc(xi)                  ## acceleration at i
    v_ver[:,:,i+1] = vi = vi + h*a      ## velocity at i-1/2 to i+1/2

E_verlet = totalEnergy(m,x_ver,0.5*(v_ver[:,:,:-1]+v_ver[:,:,1:]))

Classical Runge-Kutta 4th order

Data structures

t_RK4 = t[::4]
x_RK4 = np.zeros([len(m),3,len(t_RK4)])
v_RK4 = np.zeros([len(m),3,len(t_RK4)])
x_RK4[:,:,0] = xi = x0
v_RK4[:,:,0] = vi = v0

Time loop

for i in range(len(t_RK4)-1):
    h = t_RK4[i+1]-t_RK4[i]
    k1x = h* vi;           k1v = h*acc(xi)
    k2x = h*(vi+0.5*k1v);  k2v = h*acc(xi+0.5*k1x)
    k3x = h*(vi+0.5*k2v);  k3v = h*acc(xi+0.5*k2x)
    k4x = h*(vi+    k3v);  k4v = h*acc(xi+    k3x)
    x_RK4[:,:,i+1] = xi = xi + (k1x+2*k2x+2*k3x+k4x)/6
    v_RK4[:,:,i+1] = vi = vi + (k1v+2*k2v+2*k3v+k4v)/6

E_RK4 = totalEnergy(m,x_RK4,v_RK4)

Plots

plt.plot(t_RK4,E_RK4, t, E_verlet)
plt.legend(["RK4","Verlet"])
plt.grid(); plt.show()
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    $\begingroup$ Thanks for your answer. I completely agree with you with this passage: 'The loop order in the method step is always "outer loop - stages, inner loop - components" not "outer - components, inner - stages". ' but I don't understand where, in the code I have posted, I did that $\endgroup$
    – jack23456
    Nov 21, 2022 at 16:45
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    $\begingroup$ @jack23456 under your RK4 section you've unrolled the loop for the stages, so you handle all of body A first, then body B, then body C, but instead you must do all of stage 1 first, then all of stage 2, etc. The reason this doesn't work is because when you start to compute the forces for body A on stage 2 you're still using the positions of the other bodies before stage 1 has been computed, instead of after. $\endgroup$ Nov 21, 2022 at 16:52
  • $\begingroup$ @helloworld922 I think we are doing the following thing: At first step we compute the three components for body A, then B then C. Only at the end we update the positions. So in the next stage we use the updated positions. right? So I don't understand yourr point $\endgroup$
    – jack23456
    Nov 21, 2022 at 17:02
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    $\begingroup$ I updated my answer so hopefully it's more clear what Lutz is getting at. Let me know if that makes sense. $\endgroup$ Nov 21, 2022 at 17:29
  • $\begingroup$ @helloworld922, thank for your answers, I think I have understood the problem. I have edited the original answer, adding the code that you told me was wrong. Sorry if it is not a "clean" code. Was this the problem I was making? I plotted also the trajectories and now the collapse in the middle of the two ellipses. $\endgroup$
    – jack23456
    Nov 21, 2022 at 22:06

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