1,721,048 research outputs found
Multivalue collocation methods free from order reduction
No full textThis paper introduces multivalue collocation methods for the numerical solution of stiff problems. The presented approach does not exhibit the phenomenon of order reduction, typical of collocation based Runge–Kutta methods applied to stiff systems, since the introduced methods have uniform effective order of convergence on the overall integration interval. Examples of methods as well as numerical experiments on a selection of stiff problems are given
On the Advantages of Nonstandard Finite Difference Discretizations for Differential Problems
No full textThe goal of this work is to highlight the advantages of using NonStandard Finite Difference (NSFD) numerical schemes for the solution of ordinary differential equations (ODEs) and Partial Differential Equations (PDEs) of which some properties of the exact solution, such as positivity, are a priori known. The main reference considered is Mickens' work [14], in which the author derives NSFD schemes for ODEs and PDEs that describe real phenomena and, therefore, widely used in applications. We rigorously demonstrate that NSFD methods can have a higher order of convergence than the related classical ones, deriving also conditions that guarantee the stability of the analyzed schemes. Furthermore, we carry out in-depth numerical tests comparing classical methods with the NSFD ones proposed by Mickens, evaluating when the latter are decidedly advantageous
Nonstandard finite differences numerical methods for a vegetation reaction–diffusion model
Full text linkIn this work we derive NonStandard Finite Differences (NSFDs) (Anguelov and Lubuma, 2001; Mickens, 2020) numerical schemes to solve a model consisting of reaction-diffusion Partial Differential Equations (PDEs) that describes the coexistence of plant species in arid environments (Eigentler and Sherratt, 2019). The new methods are constructed by exploiting a-priori known properties of the exact solution, such as positivity and oscillating behavior in space. Furthermore, we extend the definition of NSFDs inspired by the Time-Accurate and High-Stable Explicit (TASE) (Bassenne et al., 2021) methodology, also exploring the existing connections between nonstandard methods and the Exponential-Fitting (EF) (Ixaru, 1997; Ixaru and Berghe, 2010) technique. Several numerical tests are performed to highlight the best properties of the new NSFDs methods compared to the related standard ones. In fact, at the same cost, the former are much more stable than the latter, and unlike them preserve all the most important features of the model
Frequency evaluation for adapted peer methods
No full textIn this paper, we consider exponentially fitted peer methods for the numerical solution of first order differential equations and we investigate how the frequencies can be tuned in order to obtain the maximal benefit. We will show that the key is analyzing the error's behavior. Formulae for optimal frequencies are computed. Numerical experiments show the properties of the proposed algorithm
Stability of two-step spline collocation methods for initial value problems for fractional differential equations
Full text linkThis paper analyzes the numerical stability of a class of two-step spline collocation methods for initial value problems for fractional differential equations. The stability region is characterized in terms of the eigenvalues of a power series, which depends on the method parameters. We provide the stability regions for several choices of the method parameters. Numerical experiments prove the effectiveness of the stability results both in a constant stepsize framework and in the case of graded meshes
Non-standard schemes for time-fractional reaction–advection–diffusion problems
No full textThe present paper concerns the numerical solution of time-fractional reaction–advection–diffusion problems. In real applications, an important issue is the preservation of qualitative properties of the analytical solution, such as positivity, which standard methods achieve only for small stepsizes. Here, novel explicit and implicit non-standard finite difference methods are introduced, by treating different terms in the approximations on different time levels, in a way to keep the solution non-negative at all times. A rigorous analysis of the stability and convergence of the proposed schemes is provided, offering robust theoretical results that illustrate their effectiveness in preserving positivity while generating accurate approximations of the solution. Finally, some numerical experiments demonstrate the efficacy of the proposed methods on different benchmark problems
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