  We develop and analyse the first second-order phase-field model to combine
melting and dissolution in multi-component flows. This provides a simple and
accurate way to simulate challenging phase-change problems in existing codes.
Phase-field models simplify computation by describing separate regions using a
smoothed phase field. The phase field eliminates the need for complicated
discretisations that track the moving phase boundary. However standard
phase-field models are only first-order accurate. They often incur an error
proportional to the thickness of the diffuse interface. We eliminate this
dominant error by developing a general framework for asymptotic analysis of
diffuse-interface methods in arbitrary geometries. With this framework we can
consistently unify previous second-order phase-field models of melting and
dissolution and the volume-penalty method for fluid-solid interaction. We
finally validate second-order convergence of our model in two comprehensive
benchmark problems using the open-source spectral code Dedalus.
