BlogMidpointMethodTimeStep.html

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  <title>Encino Sim Class : The Midpoint Method Time Step</title>

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<h2>Higher-Order Difference Equations</h2>
In our previous class, we were able to fix the instability of our spring
system by switching from the never-correct Forward Euler time step to the
Euler-Cromer time step, which made our simulation stable, but did not achieve
an accurate result over time when compared to the exact answer.
<p>
The solutions we've explored so far are primarily explicit time integrations,
in that we're only using values from previous points in time to solve for
points at future points in time. When we consider, later, implicit time
integration, where values at the current time are solved self-referentially,
things will get a lot more complex. So, we'd like to continue exploring
techniques that are fundamentally explicit in nature.
<p>
The approach that all of the higher-order explicit methods will take will be
to use lower-order approximations to compute temporary values, which will be
used to compute better derivatives that will then be used to integrate forward.
The first, and most common higher-order method is called "The Midpoint Method".

<h2>The Midpoint Method</h2>
Let's approximate the velocity at half-way through our time step, rather than
all the way to the new time, just using the forward finite-difference
approximation we used with Forward Euler:

<div class="LaTexEquation">
    \(
    a( t ) = f( t, x( t ) ) \\
    x( t + \frac{\Delta t}{2} ) \approx x( t ) + \frac{\Delta t}{2} v( t ) \\
    v( t + \Delta t ) \approx v( t ) + \Delta t a( t ) \\
    x( t + \Delta t ) \approx x( t + \frac{\Delta t}{2} ) + \frac{\Delta t}{2} v( t + \frac{\Delta t}{2} )
    \)
</div>
Now we've got one extra step of computation at each time step, computing
a midpoint position. This isn't a huge improvement over what we had before,
and here's what it looks like in code:

<pre><code>// Time Step function.
void TimeStep( float i_dt )
{
    // Compute acceleration from current position.
    float A = ( -Stiffness / BobMass ) * State[StatePositionX];

    // Update position half-way.
    State[StatePositionX] += ( i_dt/2.0 ) * State[StateVelocityX];

    // Update velocity based on acceleration.
    State[StateVelocityX] += i_dt * A;
    
    // Update position half-way.
    State[StatePositionX] += ( i_dt/2.0 ) * State[StateVelocityX];

    // Update current time.
    State[StateCurrentTime] += i_dt;
}
</code></pre>

<p>
And here that is:

<p>
<script type="application/processing" data-processing-target="pjsSpringSimp5">
float Stiffness = 5.0;
float BobMass = 0.5;
int StateSize = 3;
float[] InitState = new float[StateSize];
float[] State = new float[StateSize];
int StateCurrentTime = 0;
int StatePositionX = 1;
int StateVelocityX = 2;

int WindowWidthHeight = 300;
float WorldSize = 2.0;
float PixelsPerMeter;
float OriginPixelsX;
float OriginPixelsY;

void setup()
{
    // Create initial state.
    InitState[StateCurrentTime] = 0.0;
    InitState[StatePositionX] = 0.65;
    InitState[StateVelocityX] = 0.0;

    // Copy initial state to current state.
    // notice that this does not need to know what the meaning of the
    // state elements is, and would work regardless of the state's size.
    for ( int i = 0; i < StateSize; ++i )
    {
        State[i] = InitState[i];
    }

    // Set up normalized colors.
    colorMode( RGB, 1.0 );
    
    // Set up the stroke color and width.
    stroke( 0.0 );
    //strokeWeight( 0.01 );
    
    // Create the window size, set up the transformation variables.
    size( WindowWidthHeight, WindowWidthHeight );
    PixelsPerMeter = (( float )WindowWidthHeight ) / WorldSize;
    OriginPixelsX = 0.5 * ( float )WindowWidthHeight;
    OriginPixelsY = 0.5 * ( float )WindowWidthHeight;

    textSize( 24 );
}

// Draw our State, with the unfortunate units conversion.
void DrawState()
{
    // Compute end of arm.
    float SpringEndX = PixelsPerMeter * State[StatePositionX];

    // Compute the CORRECT position.
    float sqrtKoverM = sqrt( Stiffness / BobMass );
    float x0 = InitState[StatePositionX];
    float v0 = InitState[StateVelocityX];
    float t = State[StateCurrentTime];
    float CorrectPositionX = ( x0 * cos( sqrtKoverM * t ) ) +
        ( ( v0 / sqrtKoverM ) * sin( sqrtKoverM + t ) );
    
    // Compute draw pos for "correct"
    float CorrectEndX = PixelsPerMeter * CorrectPositionX;

    // Draw the spring.
    strokeWeight( 1.0 );
    line( 0.0, 0.0, SpringEndX, 0.0 );
          
    // Draw the spring pivot
    fill( 0.0 );
    ellipse( 0.0, 0.0, 
             PixelsPerMeter * 0.03, 
             PixelsPerMeter * 0.03 );
    
    // Draw the spring bob
    fill( 1.0, 0.0, 0.0 );
    ellipse( SpringEndX, 0.0, 
             PixelsPerMeter * 0.1, 
             PixelsPerMeter * 0.1 );

    // Draw the correct bob in blue
    fill( 0.0, 0.0, 1.0 );
    ellipse( CorrectEndX, -PixelsPerMeter * 0.25,
             PixelsPerMeter * 0.1,
             PixelsPerMeter * 0.1 );
}

// Time Step function.
void TimeStep( float i_dt )
{
    // Compute acceleration from current position.
    float A = ( -Stiffness / BobMass ) * State[StatePositionX];

    // Update position half-way.
    State[StatePositionX] += ( i_dt/2.0 ) * State[StateVelocityX];

    // Update velocity based on acceleration.
    State[StateVelocityX] += i_dt * A;
    
    // Update position half-way.
    State[StatePositionX] += ( i_dt/2.0 ) * State[StateVelocityX];

    // Update current time.
    State[StateCurrentTime] += i_dt;
}

// Processing Draw function, called every time the screen refreshes.
void draw()
{
    // Time Step.
    TimeStep( 1.0/24.0 );

    // Clear the display to a constant color.
    background( 0.75 );

    // Label.
    fill( 1.0 );
    text( "Midpoint Method", 10, 30 );

    pushMatrix();

    // Translate to the origin.
    translate( OriginPixelsX, OriginPixelsY );

    // Draw the simulation
    DrawState();
}

// Reset function. If the key 'r' is released in the display, 
// copy the initial state to the state.
void keyReleased()
{
    if ( key == 114 )
    {
        for ( int i = 0; i < StateSize; ++i )
        {
            State[i] = InitState[i];
        }
    }  
}

</script>
<canvas id="pjsSpringSimp5"> </canvas>
<p>
So really we've actually made things worse with this simple higher-order
approximation. An alternative midpoint-method approach is called
<a href="http://en.wikipedia.org/wiki/Verlet_integration#Velocity_Verlet">
Velocity Verlet</a>, and it looks like this:

<div class="LaTexEquation">
    \(
    a( t ) = f( t, x( t ) ) \\
    v( t + \frac{\Delta t}{2} ) \approx v(t) + \frac{\Delta t}{2} a( t ) \\
    x( t + \Delta t ) \approx x( t ) + \Delta t v( t + \frac{\Delta t}{2} ) \\
    a( t + \Delta t ) = f( t+\Delta t, x( t + \Delta t ) )\\
    v( t + \Delta t ) \approx v( t + \frac{\Delta t}{2} ) + \frac{\Delta t}{2} a( t + \Delta t )
    \)
</div>

And here's what that looks like in code:
<p>
<pre><code>
// Time Step function.
void TimeStep( float i_dt )
{
    // Compute acceleration from current position.
    float A = ( -Stiffness / BobMass ) * State[StatePositionX];

    // Update velocity half-way.
    State[StateVelocityX] += ( i_dt/2.0 ) * A;

    // Update position.
    State[StatePositionX] += i_dt * State[StateVelocityX];

    // Re-compute acceleration.
    A = ( -Stiffness / BobMass ) * State[StatePositionX];
    
    // Update velocity half-way.
    State[StateVelocityX] += ( i_dt/2.0 ) * A;

    // Update current time.
    State[StateCurrentTime] += i_dt;
}
</code></pre>
<p>
And here it is running:
<p>
<script type="application/processing" data-processing-target="pjsSpringSimp6">
float Stiffness = 5.0;
float BobMass = 0.5;
int StateSize = 3;
float[] InitState = new float[StateSize];
float[] State = new float[StateSize];
int StateCurrentTime = 0;
int StatePositionX = 1;
int StateVelocityX = 2;

int WindowWidthHeight = 300;
float WorldSize = 2.0;
float PixelsPerMeter;
float OriginPixelsX;
float OriginPixelsY;

void setup()
{
    // Create initial state.
    InitState[StateCurrentTime] = 0.0;
    InitState[StatePositionX] = 0.65;
    InitState[StateVelocityX] = 0.0;

    // Copy initial state to current state.
    // notice that this does not need to know what the meaning of the
    // state elements is, and would work regardless of the state's size.
    for ( int i = 0; i < StateSize; ++i )
    {
        State[i] = InitState[i];
    }

    // Set up normalized colors.
    colorMode( RGB, 1.0 );
    
    // Set up the stroke color and width.
    stroke( 0.0 );
    //strokeWeight( 0.01 );
    
    // Create the window size, set up the transformation variables.
    size( WindowWidthHeight, WindowWidthHeight );
    PixelsPerMeter = (( float )WindowWidthHeight ) / WorldSize;
    OriginPixelsX = 0.5 * ( float )WindowWidthHeight;
    OriginPixelsY = 0.5 * ( float )WindowWidthHeight;

    textSize( 24 );
}

// Draw our State, with the unfortunate units conversion.
void DrawState()
{
    // Compute end of arm.
    float SpringEndX = PixelsPerMeter * State[StatePositionX];

    // Compute the CORRECT position.
    float sqrtKoverM = sqrt( Stiffness / BobMass );
    float x0 = InitState[StatePositionX];
    float v0 = InitState[StateVelocityX];
    float t = State[StateCurrentTime];
    float CorrectPositionX = ( x0 * cos( sqrtKoverM * t ) ) +
        ( ( v0 / sqrtKoverM ) * sin( sqrtKoverM + t ) );
    
    // Compute draw pos for "correct"
    float CorrectEndX = PixelsPerMeter * CorrectPositionX;

    // Draw the spring.
    strokeWeight( 1.0 );
    line( 0.0, 0.0, SpringEndX, 0.0 );
          
    // Draw the spring pivot
    fill( 0.0 );
    ellipse( 0.0, 0.0, 
             PixelsPerMeter * 0.03, 
             PixelsPerMeter * 0.03 );
    
    // Draw the spring bob
    fill( 1.0, 0.0, 0.0 );
    ellipse( SpringEndX, 0.0, 
             PixelsPerMeter * 0.1, 
             PixelsPerMeter * 0.1 );

    // Draw the correct bob in blue
    fill( 0.0, 0.0, 1.0 );
    ellipse( CorrectEndX, -PixelsPerMeter * 0.25,
             PixelsPerMeter * 0.1,
             PixelsPerMeter * 0.1 );
}

// Time Step function.
void TimeStep( float i_dt )
{
    // Compute acceleration from current position.
    float A = ( -Stiffness / BobMass ) * State[StatePositionX];

    // Update velocity half-way.
    State[StateVelocityX] += ( i_dt/2.0 ) * A;

    // Update position.
    State[StatePositionX] += i_dt * State[StateVelocityX];

    // Re-compute acceleration.
    A = ( -Stiffness / BobMass ) * State[StatePositionX];
    
    // Update velocity half-way.
    State[StateVelocityX] += ( i_dt/2.0 ) * A;

    // Update current time.
    State[StateCurrentTime] += i_dt;
}

// Processing Draw function, called every time the screen refreshes.
void draw()
{
    // Time Step.
    TimeStep( 1.0/24.0 );

    // Clear the display to a constant color.
    background( 0.75 );

    // Label.
    fill( 1.0 );
    text( "Velocity Verlet", 10, 30 );

    pushMatrix();

    // Translate to the origin.
    translate( OriginPixelsX, OriginPixelsY );

    // Draw the simulation
    DrawState();
}

// Reset function. If the key 'r' is released in the display, 
// copy the initial state to the state.
void keyReleased()
{
    if ( key == 114 )
    {
        for ( int i = 0; i < StateSize; ++i )
        {
            State[i] = InitState[i];
        }
    }  
}

</script>
<canvas id="pjsSpringSimp6"> </canvas>
<p>
Not bad! Now we're cooking with gas! Still not perfect, though. The next class
will explore the fourth-order Runge Kutta approach, and begin to consider 
implicit methods.
</body>