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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.
Gauss–Legendre methods are implicit Runge–Kutta methods. More specifically, they are collocation methods based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method based on s points has order 2s. [1] All Gauss–Legendre methods are A-stable. [2] The Gauss–Legendre method of order two is the implicit midpoint rule.
Suppose that we want to solve the differential equation ′ = (,). The trapezoidal rule is given by the formula + = + ((,) + (+, +)), where = + is the step size. [1]This is an implicit method: the value + appears on both sides of the equation, and to actually calculate it, we have to solve an equation which will usually be nonlinear.
For example, implicit linear multistep methods include Adams-Moulton methods, and backward differentiation methods (BDF), whereas implicit Runge–Kutta methods [6] include diagonally implicit Runge–Kutta (DIRK), [7] [8] singly diagonally implicit Runge–Kutta (SDIRK), [9] and Gauss–Radau [10] (based on Gaussian quadrature [11]) numerical ...
The backward differentiation formula (BDF) is a family of implicit methods for the numerical integration of ordinary differential equations.They are linear multistep methods that, for a given function and time, approximate the derivative of that function using information from already computed time points, thereby increasing the accuracy of the approximation.
In numerical analysis and scientific computing, the backward Euler method (or implicit Euler method) is one of the most basic numerical methods for the solution of ordinary differential equations. It is similar to the (standard) Euler method , but differs in that it is an implicit method .
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin ′ ( a ) = cos( a ), meaning that the rate of change of sin( x ) at a particular angle x = a is given ...
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f ′(x + h / 2 ) and f ′(x − h / 2 ) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f: