Abstract: A computer implemented method and system for simulating a hydrocarbon reservoir. The method includes determining a computational reservoir model, comprising formation data and fluid pressure data for each of a plurality of reservoir cells, and forming a tridiagonal matrix system for each of M strongly connected lines and arranging arrays of the M tridiagonal matrix systems in a level-based data layout to be stored in a memory of a graphical processing unit (GPU). The method further includes to determining, with the GPU, an unknown potential array for each of the tridiagonal matrix systems by solving the tridiagonal matrix systems simultaneously using a Thomas method configured to operate with the level-based data layout.