Abstract
The benefits and properties of iterative splitting methods, which are based on serial versions, have been studied in recent years, this work, we extend the iterative splitting methods to novel classes of parallel versions to solve nonlinear fractional convection-diffusion equations. For such interesting partial differential examples with higher dimensional, fractional, and nonlinear terms, we could apply the parallel iterative splitting methods, which allow for accelerating the solver methods and reduce the computational time. Here, we could apply the benefits of the higher accuracy of the iterative splitting methods. We present a novel parallel iterative splitting method, which is based on the multi-splitting methods, The flexibilisation with multisplitting methods allows for decomposing large scale operator equations. In combination with iterative splitting methods, which use characteristics of waveform-relaxation (WR) methods, we could embed the relaxation behavior and deal better with the nonlinearities of the operators. We consider the convergence results of the parallel iterative splitting methods, reformulating the underlying methods with a summation of the individual convergence results of the WR methods. We discuss the numerical convergence of the serial and parallel iterative splitting methods with respect to the synchronous and asynchronous treatments. Furthermore, we present different numerical applications of fluid and phase field problems in order to validate the benefit of the parallel versions.
Highlights
Nowadays, iterative splitting methods are important solver methods to solve large systems of ordinary, partial, or stochastic differential equations, see [1,2,3,4,5,6]
Iterative splitting methods are based on two solver ideas: In the first part, we separate the full operators into different sub-operators and reduce the computational time for such sub-computation
We deal with the m-dimensional initial value problem in the non-homogeneous form, see the homogeneous form in Equation (10): c0 (t) = Ac(t) + f (t), c(0) = c0, (55)
Summary
Iterative splitting methods are important solver methods to solve large systems of ordinary, partial, or stochastic differential equations, see [1,2,3,4,5,6]. Both parts reduce the computational time and the complexity as if we solved all parts (full operator and direct method) together, see [11] Such iterative splitting methods can be used to compute, with less computational burden, an approximate solution of the ordinary differential equations (ODEs) or semi-discretized partial differential equations (PDEs), see [10,11]. We consider the interesting part of higher dimensional and nonlinear PDEs as nonlinear and fractional convection-diffusion equations, see [13,14]. We could reduce the computational cost of the time-splitting approaches, which are given as an iterative splitting approach, see [33] With such an approximation and the consideration of the semi-discretization with higher order schemes, we obtain the nonlinear differential equation system in a Cauchy-form, which is given as:.
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