Explicit and implicit methods

Explicit and implicit methods are approaches used in <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a> for obtaining numerical approximations to the solutions of time-dependent <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary</a> and <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equations</a>, as is required in <a href="/facts/Computer_simulation/Re6o8vzN">computer simulations</a> of <a href="/facts/Process_(science)/182j8RWE">physical processes</a>. Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation involving both the current state of the system and the later one. Mathematically, if 
 
 
 
 Y
 (
 t
 )
 
 
 {\displaystyle Y(t)}
 
 is the current system state and 
 
 
 
 Y
 (
 t
 +
 Δ
 t
 )
 
 
 {\displaystyle Y(t+\Delta t)}
 
 is the state at the later time (
 
 
 
 Δ
 t
 
 
 {\displaystyle \Delta t}
 
 is a small time step), then, for an explicit method

Y
        (
        t
        +
        Δ
        t
        )
        =
        F
        (
        Y
        (
        t
        )
        )
        
      
    
    {\displaystyle Y(t+\Delta t)=F(Y(t))\,}

while for an implicit method one solves an equation

G
        
          
            (
          
        
        Y
        (
        t
        )
        ,
        Y
        (
        t
        +
        Δ
        t
        )
        
          
            )
          
        
        =
        0
        
        (
        1
        )
        
      
    
    {\displaystyle G{\Big (}Y(t),Y(t+\Delta t){\Big )}=0\qquad (1)\,}

to find 
 
 
 
 Y
 (
 t
 +
 Δ
 t
 )
 .
 
 
 {\displaystyle Y(t+\Delta t).}

Explicit and implicit methods open-in-new

Explicit and implicit methods