We present a second-order finite-difference scheme for viscoelastic flows based on a recent reformulation of the constitutive laws as equations for the matrix logarithm of the conformation tensor. We present a simple analysis that clarifies how the passage to logarithmic variables remedies the high-Weissenberg numerical instability. As a stringent test, we simulate an Oldroyd-B fluid in a lid-driven cavity. The scheme is found to be stable at large values of the Weissenberg number. These results support our claim that the high Weissenberg numerical instability may be overcome by the use of logarithmic variables. Remaining issues are rather concerned with accuracy, which degrades with insufficient resolution. (c) 2005 Elsevier B.V. All rights reserved.