  We study spatially partitioned embedded Runge--Kutta (SPERK) schemes for
partial differential equations (PDEs), in which each of the component schemes
is applied over a different part of the spatial domain. Such methods may be
convenient for problems in which the smoothness of the solution or the
magnitudes of the PDE coefficients vary strongly in space. We focus on embedded
partitioned methods as they offer greater efficiency and avoid the order
reduction that may occur in non-embedded schemes. We demonstrate that the lack
of conservation in partitioned schemes can lead to non-physical effects and
propose conservative additive schemes based on partitioning the fluxes rather
than the ordinary differential equations. A variety of SPERK schemes are
presented, including an embedded pair suitable for the time evolution of
fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical
experiments are provided to support the theory.
