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For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
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:
Symbolic differentiation faces the difficulty of converting a computer program into a single mathematical expression and can lead to inefficient code. Numerical differentiation (the method of finite differences) can introduce round-off errors in the discretization process and cancellation. Both of these classical methods have problems with ...
Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes.
The mixed finite element method is a type of finite element method in which extra independent variables are introduced as nodal variables during the discretization of a partial differential equation problem.
MODFLOW simulation. MODFLOW is the U.S. Geological Survey modular finite-difference flow model, which is a computer code that solves the groundwater flow equation.The program is used by hydrogeologists to simulate the flow of groundwater through aquifers.
Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. [1] Before the emergence of computational science (also called scientific computing) as a "third way" besides theoretical and experimental sciences, computational mechanics was widely considered to be a sub-discipline of applied mechanics.
The classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if f {\displaystyle f} is a holomorphic function , real-valued on the real line, which can be evaluated at points in the complex plane near x {\displaystyle x} , then there are stable methods.