This document summarizes a project that aims to improve computational efficiency in solving large sparse linear systems for modeling ablative thermal protection system surfaces by introducing GPU techniques. Three approaches have been attempted so far: 1) using the AMGX library, 2) using new sparse matrix subclasses to perform matrix-vector products on the GPU, and 3) adjusting the CUDA_ITSOL library. The source code uses the PETSc library and takes multiple rounds to converge. Profiling found most time is spent in preconditioning and solving phases, which could benefit from GPU acceleration. The next step is to fully integrate AMGX and test different CPU-GPU ratios. Version control tools like Git are used to manage the code and track changes