I am trying to solve numerically next PDE system: $$\frac{\partial c}{\partial t}=\epsilon\frac{\partial}{\partial x}(\frac{\partial c}{\partial x}+\rho\frac{\partial \varphi}{\partial x}+\frac{vc}{1vc}\frac{\partial c}{\partial x})$$ $$\frac{\partial \rho}{\partial t}=\epsilon\frac{\partial}{\partial x}(\frac{\partial \rho}{\partial x}+c\frac{\partial \varphi}{\partial x}+\frac{v\rho}{1vc}\frac{\partial c}{\partial x})$$ $$\epsilon^2\frac{\partial \varphi^2}{\partial x^2}=\rho$$ with completely blocking boundary conditions at x= ±1; $v$ and $\epsilon$ are constants. However, at some point of calculation the oscillations near the boundaries occur due to the too big gradients near the boundaries. I have found some information that there are several techniques for handling numerical instabilities. (PetrovGalerkin (SUPG) or Galerkin leastsquares (GLS) method). Unfortunately all of them usually described for the case of simple ConvectionDiusion equation. Can anybody help me how to adjust some of those methods (or maybe another method) for my case?
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