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 nonembedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to nonphysical 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 nonoscillatory spatial discretizations. Numerical experiments are provided to support the theory.


  1. embedded Runge--Kutta methods
  2. spatially partitioned methods
  3. conservation laws
  4. method of lines

MSC codes

  1. 35L65
  2. 65L06
  3. 65M20

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Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 2887 - 2910
ISSN (online): 1095-7170


Submitted: 17 January 2013
Accepted: 24 July 2013
Published online: 30 October 2013


  1. embedded Runge--Kutta methods
  2. spatially partitioned methods
  3. conservation laws
  4. method of lines

MSC codes

  1. 35L65
  2. 65L06
  3. 65M20



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