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  2. Thermal conduction - Wikipedia

    en.wikipedia.org/wiki/Thermal_conduction

    The law of heat conduction, also known as Fourier's law (compare Fourier's heat equation), states that the rate of heat transfer through a material is proportional to the negative gradient in the temperature and to the area, at right angles to that gradient, through which the heat flows.

  3. Finite difference method - Wikipedia

    en.wikipedia.org/wiki/Finite_difference_method

    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 (+) + (() +).

  4. Crank–Nicolson method - Wikipedia

    en.wikipedia.org/wiki/Crank–Nicolson_method

    In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. [1] It is a second-order method in time. It is implicit in time, can be written as an implicit Runge–Kutta method, and it is numerically stable.

  5. FTCS scheme - Wikipedia

    en.wikipedia.org/wiki/FTCS_scheme

    In numerical analysis, the FTCS (forward time-centered space) method is a finite difference method used for numerically solving the heat equation and similar parabolic partial differential equations. [1] It is a first-order method in time, explicit in time, and is conditionally stable when applied to the heat equation.

  6. Alternating-direction implicit method - Wikipedia

    en.wikipedia.org/wiki/Alternating-direction...

    Stencil figure for the alternating direction implicit method in finite difference equations. The traditional method for solving the heat conduction equation numerically is the Crank–Nicolson method. This method results in a very complicated set of equations in multiple dimensions, which are costly to solve.

  7. Numerical solution of the convection–diffusion equation

    en.wikipedia.org/wiki/Numerical_solution_of_the...

    In this method, the basic shape function is modified to obtain the upwinding effect. This method is an extension of Runge–Kutta discontinuous for a convection-diffusion equation. For time-dependent equations, a different kind of approach is followed. The finite difference scheme has an equivalent in the finite element method (Galerkin method ...

  8. Heat equation - Wikipedia

    en.wikipedia.org/wiki/Heat_equation

    In mathematics and physics, the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. Since then, the heat equation and its variants have been found to be fundamental in ...

  9. Thermal simulations for integrated circuits - Wikipedia

    en.wikipedia.org/wiki/Thermal_simulations_for...

    The most popular methods are: Finite difference time-domain (FDTD) method, Finite element method (FEM) and method of moments (MoM). The finite-difference time-domain (FDTD) method is a robust and popular technique that consists in solving differential equations numerically as well as certain boundary conditions defined by the problem.