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i havent had a response to this question in a while, could someone please help me. Im struggling to understand the concepts of forward/backward/central differences on finite difference equations.

i have formulas for:

first order with respect to x (spatial)- forward (explicit), backward(implicit) and central

second order with respect to x- forward (explicit), backward(implicit) and central

first order with respect to t (time)- forward (explicit), backward(implicit) and central

what are the formulas for second order with respect to t- forward (explicit), backward(implicit) and central?

i have a one second order equation for time but not sure what it specifys- is it the forward (explicit)?:

$$ \frac{d^2u}{dt^2} = \frac{u_{i}(m+1)-2u_{i}(m)+u_{i}(m-1)}{(\delta t)^2} $$

i know how to solve questions just need the second order equations with respect to time. thanks in advance for any help- been looking all over the web but not really getting a grasp. thanks.

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From what I understood of your question, that is quite confusing.

A simple reference for formulas for finite differences, or their coeficientes, is found here in Wiki , for backward, forward and central.

Regarding being explicit, implicit:

  • You didn't say where you want to apply those finite difference discretizations. What is the equation, depending on what you choose backward/forward/central for each term of your equation you get a implicit or explicit schema.

    If you specify the equation will be easier, for example for the heat equation:

$$ U_t = \alpha U_{xx} $$

Using a forward difference at time and a second-order central difference for the space derivative at position $u_j$ (FTCS) we get explicit schema:

$$ \frac{u_j^{t+1}-u_j^t}{\Delta t} = \alpha \frac{u_{j+1}^t - 2u_j^t + u_{j-1}^t}{h^2} $$

where $ \Delta t $ is the discretization in time and $ h $ in space.

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