We present a new 2D finite difference code, Samovar, for high-resolution numerical modeling of complex geodynamic processes. Examples are collision of lithospheric plates (including mountain building and subduction) and lithosphere extension (including formation of sedimentary basins, regions of extended crust, and rift zones). The code models deformation of the lithosphere with viscoelastoplastic rheology, including erosion/sedimentation processes and formation of shear zones in areas of high stresses. It also models steady-state and transient conductive and advective thermal processes including partial melting and magma transport in the lithosphere. The thermal and mechanical parts of the code are tested for a series of physical problems with analytical solutions. We apply the code to geodynamic modeling by examining numerically the processes of lithosphere extension and basin formation. The results are directly applicable to the Basin and Range province, western USA, and demonstrate the roles of crust–mantle coupling, preexisting weakness zones, and erosion rate on the evolutionary trends of extending continental regions. Modeling of basin evolution indicates a critical role of syn-rift sedimentation on the basin depth and a governing role of Peierls deformation in cold lithospheric mantle. While the former may increase basin depth by 50%, the latter limits the depth of rift basins by preventing faulting in the subcrustal lithosphere.