Hi, to all experts on the Finite Volume method.
Is it absolutely essential to linearize a non-linear source ,term, of a non-linear PDE during the process of solving a PDE with finite volume method (FVM)?
Few times in textbooks and online lectures, I have saw, experts tend to linearize the non-linear source terms( for example in a 2D Poisson equation. Lets say u"-exp(u)=0 ) during the process of the discretization of a PDE with finite volume method.
However, considering the approximation of the piece-wise constant , I expect the non-linear source will add non-linear terms to discretized equations (i.e., exp( u (x_p,y_p) ), where x_p and y_p are coordination of the center of the control volume) and one can use the Newton's method to solve a set of nonlinear algebraic equation.
For example, we could have a final set of the algebraic equation as Au - diag( exp(u) )=b, where "A" is the coefficient matrix, resulting of discretization and "b" is a constant vector containing the information of boundary conditions.
What harm it could have, on the conservative nature of FVM, if one processed similar to what I have explained on above example.
Many thanks of you, if you stop by and take look at my question. :-)
Vahid
Dear Vahid,
The linearisation and the spatial discretisation are two independent aspects. To linearise or not is a decision you take regardless of the method you use, being FVM, or FEM, or DG, ...
If you have to solve a nonlinear elliptic PDE (as the one you mentioned), you have at least two options:
- Write down your linearisation scheme before discretisation; you are generating a sequence of linear elliptic equations, which you can discretise and so you end up with a sequence of linear algebraic systems.
- Discretise the given nonlinear elliptic PDE and generate in this way a nonlinear algebraic system of equations; to solve this you apply an algebraic nonlinear solver, which reduces often to solving a sequence of linear algebraic systems (as encountered e.g. when applying Newton's scheme).
In both cases, solving the linear algebraic systems generates the sequence of numerical approximations, which hopefully converges (oproving the convergence would be even nicer :))
Good luck,
Sorin
A source term in the equation is such because it acts as pointwise term and does not appear in the divergence form. Therefore, writing the volume integral of the PDE you cannot write it in the form of surface integral of the fluxes. That means you are working by definition with a PDE that does not govern a conservation of the variable.
Then, you have to analyse if the problem is stiff due to the source term. Generally you discretize in space the volume integral of the source, for time-dependent problem the time integration can be explicit or implicit (and in this case you should linearize the term).
Dear Vahid,
The linearisation and the spatial discretisation are two independent aspects. To linearise or not is a decision you take regardless of the method you use, being FVM, or FEM, or DG, ...
If you have to solve a nonlinear elliptic PDE (as the one you mentioned), you have at least two options:
- Write down your linearisation scheme before discretisation; you are generating a sequence of linear elliptic equations, which you can discretise and so you end up with a sequence of linear algebraic systems.
- Discretise the given nonlinear elliptic PDE and generate in this way a nonlinear algebraic system of equations; to solve this you apply an algebraic nonlinear solver, which reduces often to solving a sequence of linear algebraic systems (as encountered e.g. when applying Newton's scheme).
In both cases, solving the linear algebraic systems generates the sequence of numerical approximations, which hopefully converges (oproving the convergence would be even nicer :))
Good luck,
Sorin
Everything is possible but the new method will not be a finite volume method anymore.
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