Question

我一直在研究4阶Runge-Kutta求解器，并且遇到了一些困难。我已经根据文章on gafferongames编写了求解器，但是当我运行一个小的包含的例子时，我得到的错误比我用简单的Euler积分得到的差得多，即使对于简单的引力也是如此力。我把它整理成一个自包含的例子（约60行代码，包括打印），但它需要GLM才能运行。
它完整地显示了我的问题。第55行打印出分析解决方案和RK4解决方案之间的差异。这应该相对较小，但即使经过10多步之后它也会爆炸。

#include <iostream>
#include <glm/glm.hpp>
struct State{
    glm::vec3 position, velocity;
};
class Particle{
public:
    glm::vec3 position, velocity, force;
    float mass;

    void solve(float dt);
    glm::vec3 acceleration() const {return force/mass;}
    State evalDerivative(float dt, const State& curr);
    void analytic(float t,  glm::vec3 a);
};
int main(int argc, char* argv[]){
    Particle p;
    p.position = glm::vec3(0.f);
    p.mass = 1.0f;
    for(int i = 1; i <= 10; i++)    {
        p.force = glm::vec3(0.f, -9.81f, 0.f);
        p.solve(.016f);
        p.analytic(i*.016f, glm::vec3(0.f, -9.81f, 0.f));
    }
    getchar();
    return 0;
}
void Particle::solve(float dt){
    State t;t.position = glm::vec3(0.f); t.velocity = glm::vec3(0.f);
    State k1 = evalDerivative(0, t);
    State k2 = evalDerivative(dt*.5f, k1);
    State k3 = evalDerivative(dt*.5f, k2);
    State k4 = evalDerivative(dt, k3);

    position += (k1.position + 2.f*(k2.position + k3.position) + k4.position)/6.f;
    velocity += (k1.velocity + 2.f*(k2.velocity + k3.velocity) + k4.velocity)/6.f;
    force = glm::vec3(0.f);
}
State Particle::evalDerivative(float dt, const State& curr){
    State s;
    s.position = position + curr.position*dt;
    s.velocity = velocity + curr.velocity*dt;

    s.position = s.velocity;
    s.velocity = acceleration();
    return s;
}
void Particle::analytic(float t, glm::vec3 a){
    glm::vec3 tPos = glm::vec3(0.f) + 0.5f*a*t*t;
    glm::vec3 tVel = glm::vec3(0.f) + a*t;

    glm::vec3 posdiff = tPos - position;
    glm::vec3 veldiff = tVel - velocity;
    std::cout << "POSITION: " << posdiff.x << ' ' << posdiff.y << ' ' << posdiff.z << std::endl;
    std::cout << "VELOCITY: " << veldiff.x << ' ' << veldiff.y << ' ' << veldiff.z << std::endl << std::endl;
}

如果有人能帮到我这里，我就在这个系绳的尽头。

Answer 1

嗯，我觉得很蠢。我已经在这个工作了几个小时了，我错过了一小步：

position += (k1.position + 2.f*(k2.position + k3.position) + k4.position)/6.f;
velocity += (k1.velocity + 2.f*(k2.velocity + k3.velocity) + k4.velocity)/6.f;

应该是

position += (k1.position + 2.f*(k2.position + k3.position) + k4.position)*dt/6.f;
velocity += (k1.velocity + 2.f*(k2.velocity + k3.velocity) + k4.velocity)*dt/6.f;

Runge-Kutta四阶积分器出错

1 个答案: