I'm trying to implement a collision detection system, and it is working for the most part, no overlapping (or at most very little overlapping) of characters, and wall collisions. The problem is that i have a bunch of characters following a player and just run into it, and when there are about 15-20 of those characters all pushing at the player, it can lead to the player or other objects being pushed through walls.

My code works as follows, first I update all of the characters, and they check collisions against each other, then I check for any character collisions with the walls. I think the problem is that I'm only ever checking one object against another, where as i need to be able to correctly detect multiple simultaneous collisions.

I've been looking into speculative contact and continuous collisions, but unfortunately i start to get lost when the non-code math equations come up. I don't necessarily need a real code example, but a good pseudo code/explanation would be great.

Code below if necessary, although a thorough explanation of how to mend this is also sufficient.

Character update/collisions:

void CharacterManager::updateAll(float elapsedTime)
{
    for(std::vector<std::shared_ptr<Character>>::iterator i = _characters.begin(); i != _characters.end(); i++) {
        (*i)->update(elapsedTime);
    }
    collisions();
}

void CharacterManager::collisions()
{
    for(std::vector<std::shared_ptr<Character>>::iterator i = _characters.begin(); i != _characters.end(); i++) {
        for(std::vector<std::shared_ptr<Character>>::iterator j = _characters.begin(); j != _characters.end(); j++) {
            if(i == j) continue;
            float xi = (*i)->position().x;
            float yi = (*i)->position().y;
            float xj = (*j)->position().x;
            float yj = (*j)->position().y;
            float dx = xi - xj;
            float dy = yi - yj;
            float distSquared = dx * dx + dy * dy;
            float ri = (*i)->xRect().width/2;
            float rj = (*j)->xRect().width/2;
            if(distSquared < (ri + rj) * (ri + rj)) {
                // fix collisions
                float angle = atan2f(dy,dx);
                float overlap = (ri + rj) - sqrt(distSquared);
                if(xi < xj) {
                    if(yi < yj) {
                        (*i)->position(xi - cosf(angle) * overlap/2, yi - sinf(angle) * overlap/2);
                        (*j)->position(xj + cosf(angle) * overlap/2, yj + sinf(angle) * overlap/2);
                    } else {
                        (*i)->position(xi - cosf(angle) * overlap/2, yi + sinf(angle) * overlap/2);
                        (*j)->position(xj + cosf(angle) * overlap/2, yj - sinf(angle) * overlap/2);
                    }
                } else {
                    if(yi < yj) {
                        (*i)->position(xi + cosf(angle) * overlap/2, yi - sinf(angle) * overlap/2);
                        (*j)->position(xj - cosf(angle) * overlap/2, yj + sinf(angle) * overlap/2);
                    } else {
                        (*i)->position(xi + cosf(angle) * overlap/2, yi + sinf(angle) * overlap/2);
                        (*j)->position(xj - cosf(angle) * overlap/2, yj - sinf(angle) * overlap/2);
                    }
                }
                // calc new velocities
                float vxi = (*i)->velocity().x;
                float vyi = (*i)->velocity().y;
                float vxj = (*j)->velocity().x;
                float vyj = (*j)->velocity().y;
                float vx = vxj - vxi;
                float vy = vyj - vyi;
                float dotProduct = dx * vx + dy * vy;
                if(dotProduct >= 0) {

                    float collisionScale = dotProduct / distSquared;
                    float xCollision = dx * collisionScale;
                    float yCollision = dy * collisionScale;
                    float combinedMass = (*i)->weight() + (*j)->weight();
                    float collisionWeightA = 2 * (*j)->weight() / combinedMass;
                    float collisionWeightB = 2 * (*i)->weight() / combinedMass;
                    (*i)->velocity(vxi + collisionWeightA * xCollision, vyi + collisionWeightA * yCollision);
                    (*j)->velocity(vxj - collisionWeightB * xCollision, vyj - collisionWeightB * yCollision);
                }
            }
        }
    }
}

Wall collisions:

void Stage::characterCrossCollisions(std::shared_ptr<Character> character)
{
    for(std::vector<std::shared_ptr<Tile>>::iterator tile = tiles.begin(); tile != tiles.end(); tile++) {
        if(!(*tile)->walkable) {
            sf::Rect<float> cxr = character->xRect();
            sf::Rect<float> cyr = character->yRect();
            sf::Rect<float> tr = (*tile)->getRect();

            if(!(cxr.left > tr.left + tr.width ||
                 cxr.left + cxr.width < tr.left ||
                 cxr.top > tr.top + tr.height ||
                 cxr.top + cxr.height < tr.top)) {
                float ox = 0;
                if(character->position().x > (*tile)->position().x) {
                    ox = cxr.left - (tr.left + tr.width);
                }
                else {
                    ox = cxr.left + cxr.width - tr.left;
                }
                character->position(character->position().x - ox, character->position().y);
            }

            if(!(cyr.left > tr.left + tr.width ||
                 cyr.left + cyr.width < tr.left ||
                 cyr.top > tr.top + tr.height ||
                 cyr.top + cyr.height < tr.top)) {
                float oy = 0;
                if(character->position().y > (*tile)->position().y) {
                    oy = cyr.top - (tr.top + tr.height);
                }
                else {
                    oy = cyr.top + cyr.height - tr.top;
                }
                character->position(character->position().x, character->position().y - oy);
            }
        }
    }
}