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| | Oscar is how he's called and he totally loves this name. He utilized to be unemployed but now he is a meter reader. One of the extremely very best issues in the globe for me is to do aerobics and I've been performing it for fairly a while. For a whilst I've been in South Dakota and my mothers and fathers reside close by.<br><br>Here is my web blog ... [http://www.streaming.iwarrior.net/blog/396549 http://www.streaming.iwarrior.net/] |
| '''Explicit and implicit methods''' are approaches used in [[numerical analysis]] for obtaining numerical solutions of time-dependent [[ordinary differential equation|ordinary]] and [[partial differential equation]]s, as is required in [[computer simulation]]s of [[Process (science)|physical processes]]. These form a part of [[temporal discretization]] carried out; together with [[Crank–Nicolson method]]. | |
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| '''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 <math>Y(t)</math> is the current system state and <math>Y(t+\Delta t)</math> is the state at the later time (<math>\Delta t</math> is a small time step), then, for an explicit method
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| : <math>Y(t+\Delta t) = F(Y(t))\,</math>
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| while for an implicit method one solves an equation | |
| : <math>G(Y(t), Y(t+\Delta t))=0 \quad\quad (1)\,</math>
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| to find <math>Y(t+\Delta t).</math>
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| It is clear that implicit methods require an extra computation (solving the above equation), and they can be much harder to implement. Implicit methods are used because many problems arising in practice are [[Stiff equation|stiff]], for which the use of an explicit method requires impractically small time steps <math>\Delta t</math> to keep the error in the result bounded (see [[numerical stability]]). For such problems, to achieve given accuracy, it takes much less computational time to use an implicit method with larger time steps, even taking into account that one needs to solve an equation of the form (1) at each time step. That said, whether one should use an explicit or implicit method depends upon the problem to be solved.
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| ==Illustration using the forward and backward Euler methods==
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| Consider the [[ordinary differential equation]]
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| : <math>\frac{dy}{dt} = -y^2, \ y\in [0, a]\quad \quad (2)</math> | |
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| with the initial condition <math>y(0)=1.</math> Consider a grid <math>t_k=a\frac{k}{n}</math> for 0≤''k''≤''n'', that is, the time step is <math>\Delta t=a/n,</math> and denote <math>y_k=y(t_k)</math> for each <math>k</math>. [[Discretization|Discretize]] this equation using the simplest explicit and implicit methods, which are the ''forward Euler'' and ''backward Euler '' methods (see [[numerical ordinary differential equations]]) and compare the obtained schemes.
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| ;Forward Euler method:
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| The forward [[Euler method]]
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| :<math>\left(\frac{dy}{dt}\right)_k = \frac{y_{k+1}-y_k}{\Delta t} = - y_k^2</math>
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| yields
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| : <math>y_{k+1}=y_k-\Delta t y_k^2 \quad \quad \quad(3)\, </math>
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| for each <math>k=0, 1, \dots, n.</math> This is an explicit formula for <math>y_{k+1}</math>.
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| ;Backward Euler method:
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| With the [[backward Euler method]]
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| :<math>\frac{y_{k+1}-y_k}{\Delta t} = - y_{k+1}^2</math>
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| one finds the implicit equation
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| : <math>y_{k+1}+\Delta t y_{k+1}^2=y_k</math> | |
| for <math>y_{k+1}</math> (compare this with formula (3) where <math>y_{k+1}</math> was given explicitly rather than as an unknown in an equation).
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| This is a [[quadratic equation]], having one negative and one positive [[Root of a function|root]]. The positive root is picked because in the original equation the initial condition is positive, and then <math>y</math> at the next time step is given by
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| : <math>y_{k+1}=\frac{-1+\sqrt{1+4\Delta t y_k}}{2 \Delta t}. \quad \quad (4)</math>
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| In the vast majority of cases, the equation to be solved when using an implicit scheme is much more complicated than a quadratic equation, and no exact solution exists. Then one uses [[root-finding algorithm]]s, such as [[Newton's method]].
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| ==See also==
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| * [[Courant–Friedrichs–Lewy condition]]
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| * [[SIMPLE algorithm]], a semi-implicit method for pressure-linked equations
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| {{DEFAULTSORT:Explicit And Implicit Methods}}
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| [[Category:Numerical differential equations]]
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Here is my web blog ... http://www.streaming.iwarrior.net/