Argonne’s Mathematics and Computer Science Division is developing scalable numerical methods encapsulated in robust software that enables scientific simulation on extreme-scale computers. Our research integrates work with applied mathematicians, computer scientists, and domain scientists both across the laboratory and throughout the world, developing the tools essential for new scientific discoveries and solutions to complex, real-world problems. Our strengths include computational fluid dynamics, higher-order methods, meshing techniques, and multiphysics software. Of particular note is our work in the U.S. Department of Energy’s SciDAC (Scientific Discovery through Advanced Computing) program and the Exascale Computing Program, through which we exploit high-performance computing and beyond-Moore computing, applying the most advanced computer science and mathematics techniques to DOE’s most critical applications in diverse areas such as astrophysics, Earth systems, energy systems, engineering diagnostics, materials science, and nuclear reactor simulation.