This chapter starts with conservation laws and their numerical methods, which are the bases for the simulation of compressible flows. It presents concepts such as weak solution and conservation error, and discusses classical conservative schemes. Then, it proceeds to compute the conservation laws via domain decomposition. It discusses conservative interface algorithms and proves that a converged numerical solution is a weak solution of a conservation law. Numerical examples demonstrate the advantages of conservative interface algorithms in correctly capturing shock location and strength. Analysis indicates that the conservative interface algorithm built by directly enforcing numerical fluxes that contain grid spacing and/or time step suffers from problems in two scenarios: it becomes inconsistent with the conservation law when the subdomains have different grid spacings and when they use different discretization schemes. Following these, the chapter deals with the computation with nonconservative interface algorithms. It analyzes the conservation error resulting from a nonconservative interface treatment. Theoretical analysis explains why a nonconservative interface algorithm may work; it proves that the converged solution associated with the nonconservative interface algorithm is a weak solution under certain conditions, followed by illustrative numerical examples. Additionally, a counterexample illustrates that a nonconservative interface algorithm can fail.