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Spatially partitioned embedded Runge-Kutta Methods
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
Numerical Methods for Solving Convection-Diffusion Problems
Convection-diffusion equations provide the basis for describing heat and mass
transfer phenomena as well as processes of continuum mechanics. To handle flows
in porous media, the fundamental issue is to model correctly the convective
transport of individual phases. Moreover, for compressible media, the pressure
equation itself is just a time-dependent convection-diffusion equation.
For different problems, a convection-diffusion equation may be be written in
various forms. The most popular formulation of convective transport employs the
divergent (conservative) form. In some cases, the nondivergent (characteristic)
form seems to be preferable. The so-called skew-symmetric form of convective
transport operators that is the half-sum of the operators in the divergent and
nondivergent forms is of great interest in some applications.
Here we discuss the basic classes of discretization in space: finite
difference schemes on rectangular grids, approximations on general polyhedra
(the finite volume method), and finite element procedures. The key properties
of discrete operators are studied for convective and diffusive transport. We
emphasize the problems of constructing approximations for convection and
diffusion operators that satisfy the maximum principle at the discrete level
--- they are called monotone approximations.
Two- and three-level schemes are investigated for transient problems.
Unconditionally stable explicit-implicit schemes are developed for
convection-diffusion problems. Stability conditions are obtained both in
finite-dimensional Hilbert spaces and in Banach spaces depending on the form in
which the convection-diffusion equation is written
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