# Questions tagged [nonlinear-equations]

Solution of nonlinear systems of equations. The equations might be algebraic or differential equations.

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### How to discretize a non-linear PDE with boundary conditions and intial value

Consider this non linear PDE: $$u_t + c(u^2)_x =\alpha u_{xx} \text{ , } -1<x<1 , t>0$$ with $$u(-1,t) = g_L(t) \text{ , } u(1,t) = g_R(t) \text{ and } u(x,0)=f(x)$$ where the 3 functions(...
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### How does the error work for the Strang Splitting?

We know in Strang splitting that the splitting error in the steady state solution is proportional to $h^2$. I want ask 2 things: If this error in the steady state solution is the global error? If we ...
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### Discretization of a non-linear ODE using FDM isn't grid indepenent

I am trying to solve the ODE : $\frac{d^2T}{dx^2} = \omega_1 T+\omega_2 T^2$ + using different numerical methods. I have tried the following discretizations so far and none of them seem to be grid ...
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### Derivative-free ill-conditioned non-linear least squares

I am looking for a package which can solve (non-linear) least squares problems without the use of derivatives (because of an expensive model), but which also deals with ill-conditioning well (such as ...
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### Jacobians with automatic differentiation

I have an objective function F: Nx1 -> Nx1, where N>30000. There are many sparse matrix/tensor multiplications in this function, so taking an analytic Jacobian by paper and pen is cumbersome. ...
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### How avoid square shape with Laplacian operator in reaction diffusion calculations?

I have used different variants of the Laplacian operator (div grad) using 4, 8, 12, 20 and 24 of the closest points. I get problems due to the chosen coordinate system and the discretization of the ...
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### RK4-method starts oscillating above certain input parameters

I am trying to solve an equation of the following type $$\partial_zE(z)=-c_0J$$ with $$J=c_1\beta E^3(z)$$ using the boost::odeint-framework and a fixed time stepper, with $c_0$, $c_1$ and $\beta$ ...
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### Issue solving nonlinear equation containing a quotient

I have a coupled set of PDEs that need to be solved as part of a larger problem. I am currently approaching this by computing spatial derivatives with finite differences and using PETSc's nonlinear ...
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### Methods for solving discrete PDEs using algorithmic differentiation results

I'm looking for a method to solve a 20000 variable, 20000 residual non-linear PDE with a Galerkin method. I have Fortran subroutines for: The residuals: $\vec{r}(\vec{x})$; Their Jacobian multiplied ...
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I want to estimated the parameters $\ \hat{\theta}$ of a model using an iterative search for the minimum of a cost function. The cost function is defined as follows: $$V_N(\hat{\theta}) = \frac{1}{... 0answers 90 views ### Linearize non-linear PDE with BCs to hyperbolic problem: How does linearization affect BCs? I am working with the Shallow Water equations that is a system of non-linear PDEs that simulate water waves propagation on some domain, in my cases the x-axis. I have here a GIF showing the results (... 1answer 94 views ### Full approximation scheme - smoothers - literature recomendation I would like to use full approximation scheme(FAS) for solving nonlinear PDE. I am looking into practical implementation of FAS in parallel(MPI) settings. I noticed the most common/effective smoother ... 0answers 75 views ### Solving a nonlinear problem with a very small components with finite element method In solving nonlinear hyperelastic solid mechanics problems, to converge to the correct solution we need to do step-by-step loading which makes the deformation at each step very small (for my ... 1answer 71 views ### Solve convection-diffusion equation with a non-linear source term I would like to solve this equation (which is adapted in my case, a plug flow reactor when a reaction occurs):  \frac{df}{dt} = D \frac{d²f}{dz²} - u \frac{df}{dz} + r(z,t)  with r(z,t)= - k f^{n}... 0answers 123 views ### Incorporating radiation boundary condition at the edge in finite difference I am trying to solve the 2-d heat equation on a rectangle using finite difference method. I am confused as to how to incorporate non linear radiation boundary condition at the edge. -k\frac{\partial ... 1answer 125 views ### Numerically solving nonlinear parabolic stochastic PDEs For a project I'm doing, I have to numerically solve a nonlinear parabolic stochastic partial differential equation, of the form$$ u_t = u_{xx} + f(u)(u_x)^2 + a(u) + b(u)W(t, x), $$where primes ... 1answer 51 views ### Nonlinear least squares resolution matrix For a linear least squares problem, it is possible to define a resolution matrix, relating the estimated model parameters to the true model parameters. If we are solving a regularized problem,$$ \...
For a project in my advanced numerical method class I have to solve the 1D Kuramoto-Sivashinsky equation. $$u_t + u u_x + \lambda u_{xx} + \eta u_{xxxx} = 0.$$ As explained here I will solve it ...