Restoration of the contact surface in FORCE-type centred schemes II: Non-conservative one- and two-layer two-dimensional shallow water equations

► A modification of the FORCE′ scheme for non-conservative systems (named FORCE′-Contact) is presented. ► The scheme is applied to non-conservative one-layer and two-layer shallow water equations. ► The new scheme, unlike the original, is well balanced for a suspended tracer. ► The new scheme performs better than the original for contact waves and steady states in meandering rivers.