I am trying to solve finite difference heat equation by Newton–Raphson method
There are two ways to solve Non-linear PDE/ODE. First quasilinearize the non-linear term(s) in the equation and then take approximation. Solve the corresponding linear system to get solution or first take approximation to get a system of non-linear equations and solve them by Newton-Raphson method. At each iteration find the value of Jacobian and find new refined value till accuracy is satisfied.
There are two ways to solve Non-linear PDE/ODE. First quasilinearize the non-linear term(s) in the equation and then take approximation. Solve the corresponding linear system to get solution or first take approximation to get a system of non-linear equations and solve them by Newton-Raphson method. At each iteration find the value of Jacobian and find new refined value till accuracy is satisfied.
I am doing the same thing but not getting the results. Actually i am trying to find the time propagation of shock wave (step).
The heat equation is parabolic. No pb then, even if the loading is given in term of steps. You may simply have overshoots.
There are many publications on the subject. You must choose a method for time integration : explicit, semi -implicit, or implicit , one step or multi-step , first-order (Euler ) , second-order or higher-order (Runge-Kutta ), ... .
It is only once you have chosen the integration time scheme that you choose your solver. If the scheme is explicit, there is just a linear system to solveat each time step: it can be with a direct solver ( Gauss pivot type) or with an iterative solver ( Gauss -Seidel ) . If the scheme is semi -implicit or implicit, you must solve a non-linear system at each time step.
Many methods can be used: fixed-point , Newton-Raphson , quasi-Newton, modified Newton, among others ...
To solve the heat equation , I use a semi -implicit second-order scheme with a fixed point method or an explicit Runge-Kutta RK4 scheme, with a direct solver if the number of unknown is less than 100,000 and with a Gauss-Seidel solver otherwise.
It is not a good idea to use the NR method because it has the problem of starting point to converge.
There several finite difference schemes which are able to solve your popular equation.
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