Despite great advances in stabilizing the numerical simulation of viscoelastic fluid flows at high Weissenberg numbers, this remains a major challenge for researchers. The breakdown of numerical algorithms, known as HWNP (High Weissenberg Number Problem), has often been associated with steep stress gradients in narrow regions of the flow domain. One of the consequences of the HWNP is the loss of the positive definiteness of the symmetric molecular strain tensor at the continuum level, which is in viscoelastic fluids represented by the conformation tensor. To stabilize the numerical simulation, algorithms often focus on introducing accurate and smooth interpolation of velocity gradients in the solution of the constitutive equation. In this work, numerical tests are presented, demonstrating the HWNP problem and its possible cure based on the stabilization method employing addition of a local (in space) artificial stress diffusion term to the transport equations. The problem and corresponding solution methods are demonstrated on a case of flow in a two-dimensional corrugated channel for a range of Weissenberg number, showing where and how the positive definiteness of the conformation tensor can be violated.