Artificial boundary conditions for diffusion equations: Numerical study
The paper is devoted to the numerical study of the advection-diffusion equation \[ \partial_t u+ (v\cdot \nabla) u-\nu \Delta u=0, \] where \(\nu\) is the viscosity (constant) and \(v= (v_1, v_2)\) is the advection velocity. In order to solve this equation in the disk \(D(R)= \{|X |< R\}\) with initial data having a compact support in \(D(R)\) the author considers the above equation in polar coordinates \(r,\theta\). Then \(\partial_ru\) on \(\partial D\) are determined by \(u\) on \(\partial D\) by the Dirichlet-to-Neumann operator \(N\), namely \(\partial_r u= -Nu\) (so-called transparent boundary condition). In order to compute \(N\) the author studies the symbol of \(N\) and approximates locally the operators represented in it. For doing this it is necessary to interpolate the function \(z^{1/2}\) by rational fractions in \(\mathbb{C} \setminus(-\infty, 0)\). Introducing suitably choosen interpolation points the author obtains explicitly the artificial boundary condition on \(\partial D\). Next, he takes \(v=0\), the Gaussian curve as initial data and \(u= u(t,r)\) for \(R\geq 2\). (Now interpolation points are choosen on the imaginary axis). He then tests the method numerically obtaining good results (compared with the exact solution) in the disk \(D(2)\). In the last section the author comes back to the case when \(v\) is not zero with \(u(X,0)= \exp (-\gamma |X |^2)\) \((\gamma>0)\) and again checks numerically the method.
scientific article; zbMATH DE number 908434