High Intensity Focused Ultrasound (HIFU) is a non-invasive ablative therapy. It is usually performed under MR monitoring, which provides reliable real-time thermal information to ensure a complete tumor ablation while preserving as much healthy tissue as possible. Unfortunately, many patients do not necessarily have access to this expensive and cumbersome cutting-edge technology, which is prohibitive for a widespread use of MRI to guide thermal ablation procedures. Ultrasound (US) is a promising low cost and portable alternative, that allows real-time monitoring and can easily be deployed outside hospitals. However, US-based thermometry alone is not robust enough for the monitoring of in-vivo tissue ablation, and its feasibility is demonstrated only on in-vitro cases for small range of temperatures, up to 50°C. Computational models can simulate the biophysical phenomena and mechanisms which govern this complex thermal therapy. The US wave propagation, the temperature evolution as well as the resulted necrotic lesion can be modeled. A method integrating those sources of information to intra-operative US data would allow to recover the accurate temperature in a wider range. Therefore, US thermometry could be improved and provide an inexpensive yet comprehensive method for intra-procedural monitoring of the ablative process through HIFU. In this paper, we propose to study the rise in temperature induced by high intensity US propagation in biological tissue, which is particularly difficult to simulate due to the complexity of the involved phenomena. The physics-based HIFU model simulates the nonlinear US propagation using a k-space model coupled with the heat propagation in biological tissue using a reaction-diffusion equation. We analyze numerically the model to evaluate its accuracy and related computational cost. Finally, our simulation approach is validated against MR thermometry, the gold-standard monitoring tool used in clinical setting. Three consecutive HIFU ablations were performed on a 2% agar and 2% silicon phantom using the Sonalleve V2 MR-HIFU system (Profound Medical, Toronto, Canada).