java - Runge-Kutta N-Body 实现不起作用

Question

我已经按照本文所述用 Java 实现了 Runge-Kutta 4 算法。 http://spiff.rit.edu/richmond/nbody/OrbitRungeKutta4.pdf

然而，运动是不稳定的。即使只有两个 body ，也没有周期性运动。

protected void integrateRK4(double time) {

    final double H = time;
    final double HO2 = H/2;
    final double HO6 = H/6;

    Vector2[] currentVelocities = new Vector2[objects.length];
    Vector2[] currentPositions = new Vector2[objects.length];
    Vector2[] vk1 = new Vector2[objects.length];
    Vector2[] vk2 = new Vector2[objects.length];
    Vector2[] vk3 = new Vector2[objects.length];
    Vector2[] vk4 = new Vector2[objects.length];
    Vector2[] rk1 = new Vector2[objects.length];
    Vector2[] rk2 = new Vector2[objects.length];
    Vector2[] rk3 = new Vector2[objects.length];
    Vector2[] rk4 = new Vector2[objects.length];


    for (int i=0; i<objects.length; i++) {
        currentVelocities[i] = objects[i].velocity().clone();
        currentPositions[i] = objects[i].position().clone();
    }

        vk1 = computeAccelerations(objects);
    for (int i=0; i<objects.length; i++) {
        rk1[i] = currentVelocities[i].clone();
    }

    for (int i=0; i<objects.length; i++) {
        objects[i].setPosition(Vectors2.add(currentPositions[i], Vectors2.prod(rk1[i], HO2)));
    }
        vk2 = computeAccelerations(objects);

    for (int i=0; i<objects.length; i++) {
        rk2[i] = Vectors2.prod(vk1[i], HO2);
    }


    for (int i=0; i<objects.length; i++) {
        objects[i].setPosition(Vectors2.add(currentPositions[i], Vectors2.prod(rk2[i], HO2)));
    }
        vk3 = computeAccelerations(objects);

    for (int i=0; i<objects.length; i++) {
        rk3[i] = Vectors2.prod(vk2[i], HO2);
    }


    for (int i=0; i<objects.length; i++) {
        objects[i].setPosition(Vectors2.add(currentPositions[i], Vectors2.prod(rk3[i], H)));
    }
        vk4 = computeAccelerations(objects);

    for (int i=0; i<objects.length; i++) {
        rk4[i] = Vectors2.prod(vk3[i], H);
    }


    for (int i=0; i<objects.length; i++) {
        objects[i].setVelocity(Vectors2.add(currentVelocities[i], Vectors2.prod(Vectors2.add(vk1[i], Vectors2.prod(vk2[i], 2), Vectors2.prod(vk3[i], 2), vk4[i]), HO6)));
        objects[i].setPosition(Vectors2.add(currentPositions[i], Vectors2.prod(Vectors2.add(rk1[i], Vectors2.prod(rk2[i], 2), Vectors2.prod(rk3[i], 2), rk4[i]), HO6)));
    }
}

我的实现不知何故不正确吗？

注意:

Vectors2 是我自己实现的 Vectors，它是大小为 2 的一阶张量。
所有静态方法 Vectors2.* 都返回 vector 的副本。
所有在 Vector2 实例上调用的非静态方法都会修改 vector ，同样适用于 objects[i].addVelocity() 和 objects[i].addPosition()

Vectors2.add(Vector2... vectors2) 进行逐元素加法。
Vectors2.prod(Vector2... vectors2) 执行逐元素乘法。
Vectors2.prod(Vector2 vector2, double c) 执行逐元素乘法。

computeAccelerations(PointBody[] objects) 返回加速度 vector 数组。
变量 objects 是类 NBodyIntegrator 的一个属性，这些函数是其中的一部分。它是一个 PointBody[] 数组。

为了确保这不是其他错误，我通过删除 k2、k3 和 k4 将 RK4 算法缩减为显式欧拉算法。这一个按预期工作。

protected void integrateRK1(double time) {
    final double H = time;
    final double HO2 = H/2;

    Vector2[] currentVelocities = new Vector2[objects.length];
    Vector2[] currentPositions = new Vector2[objects.length];
    Vector2[] vk1;
    Vector2[] rk1 = new Vector2[objects.length];


    for (int i=0; i<objects.length; i++) {
        currentVelocities[i] = objects[i].velocity().clone();
        currentPositions[i] = objects[i].position().clone();
    }


        vk1 = computeAccelerations(objects);
    for (int i=0; i<objects.length; i++) {
        rk1[i] = currentVelocities[i].clone();
    }


    for (int i=0; i<objects.length; i++) {
        objects[i].setVelocity(Vectors2.add(currentVelocities[i], Vectors2.prod(Vectors2.add(vk1[i]), H)));
        objects[i].setPosition(Vectors2.add(currentPositions[i], Vectors2.prod(Vectors2.add(rk1[i]), H)));
    }
}

Answer 1

你在设置

rk1 = v0
pos2 = pos0 + rk1*h/2
vk2 = acc(pos2)

这是正确的。但是然后你继续

rk2 = vk1*h/2

它应该在哪里

rk2 = v0 + vk1*h/2

等等。当然那么累积位置更新也是错误的。