1,721,302 research outputs found
Numerical approximation of nonlinear BVPs by means of BVMs
No full textBoundary Value Methods (BVMs) would seem to be suitable candidates for
the solution of nonlinear Boundary Value Problems (BVPs). They have been
successfully used for solving linear BVPs together with a mesh selection
strategy based on the conditioning of the linear systems. Our aim is to
extend this approach so as to use them for the numerical approximation
of nonlinear problems. For this reason, we consider the
quasi-linearization technique that is an application of the Newton
method to the nonlinear differential equation. Consequently, each
iteration requires the solution of a linear BVP. In order to guarantee
the convergence to the solution of the continuous nonlinear problem, it
is necessary to determine how accurately the linear BVPs must be solved.
For this goal, suitable stopping criteria on the residual and on the
error for each linear BVP are given. Numerical experiments on stiff
problems give rather satisfactory results, showing that the experimental
code, called TOM, that uses a class of BVMs and the quasi-linearization
technique, may be competitive with well known solvers for BVPs
High order generalized upwind schemes and numerical solution of singular perturbation problems
No full textHigh even order generalizations of the traditional upwind method are introduced to solve second order ODE-BVPs without recasting the problem as a first order system. Both theoretical analysis and numerical comparison with central difference schemes of the same order show that these new methods may avoid typical oscillations and achieve high accuracy. Singular perturbation problems are taken into account to emphasize the main features of the proposed methods
High Order Generalized Upwind Schemes and the Numerical Solution of Singular Perturbation Problems
No full textHigh order finite difference schemes for the solution of second order BVPs
No full textWe introduce new methods in the class of boundary value methods (BVMs) to solve boundary value problems (BVPs) for a second-order ODE.These formulae correspond to the high-order generalizations of classical finite difference schemes for the first and second derivatives.In this research, we carry out the analysis of the conditioning and of the time-reversal symmetry of the discrete solution for a linear convection–diffusion ODE problem. We present numerical examples emphasizing the good convergence behavior of the new schemes. Finally, we show how these methods can be applied in several space dimensions on a uniform mesh
Matrix-oriented discretization methods for reaction–diffusion PDEs: Comparisons and applications
Full text linkSystems of reaction-diffusion partial differential equations (RD-PDEs) are widely applied for modeling life science and physico-chemical phenomena. In particular, the coupling between diffusion and nonlinear kinetics can lead to the so-called Turing instability, giving rise to a variety of spatial patterns (like labyrinths, spots, stripes, etc.) attained as steady state solutions for large time intervals. To capture the morphological peculiarities of the pattern itself, a very fine space discretization may be required, limiting the use of standard (vector-based) ODE solvers in time because of excessive computational costs. By exploiting the structure of the diffusion matrix, we show that matrix-based versions of time integrators, such as Implicit–Explicit (IMEX) and exponential schemes, allow for much finer problem discretizations. We illustrate our findings by numerically solving the Schnakenberg model, prototype of RD-PDE systems with Turing pattern solutions, and the DIB-morphochemical model describing metal growth during battery charging processes
Bulk-surface virtual element method for systems of PDEs in two-space dimensions
Full text linkIn this paper we consider a coupled bulk-surface PDE in two space dimensions. The model consists of a PDE in the bulk that is coupled to another PDE on the surface through general nonlinear boundary conditions. For such a system we propose a novel method, based on coupling a virtual element method (Beirão Da Veiga et al. in Math Models Methods Appl Sci 23(01):199–214, 2013. https://doi.org/10.1051/m2an/2013138) in the bulk domain to a surface finite element method (Dziuk and Elliott in Acta Numer 22:289–396, 2013. https://doi.org/10.1017/s0962492913000056) on the surface. The proposed method, which we coin the bulk-surface virtual element method includes, as a special case, the bulk-surface finite element method (BSFEM) on triangular meshes (Madzvamuse and Chung in Finite Elem Anal Des 108:9–21, 2016. https://doi.org/10.1016/j.finel.2015.09.002). The method exhibits second-order convergence in space, provided the exact solution is H2 + 1 / 4 in the bulk and H2 on the surface, where the additional 14 is required only in the simultaneous presence of surface curvature and non-triangular elements. Two novel techniques introduced in our analysis are (i) an L2-preserving inverse trace operator for the analysis of boundary conditions and (ii) the Sobolev extension as a replacement of the lifting operator (Elliott and Ranner in IMA J Num Anal 33(2):377–402, 2013. https://doi.org/10.1093/imanum/drs022) for sufficiently smooth exact solutions. The generality of the polygonal mesh can be exploited to optimize the computational time of matrix assembly. The method takes an optimised matrix-vector form that also simplifies the known special case of BSFEM on triangular meshes (Madzvamuse and Chung 2016). Three numerical examples illustrate our findings
A unifying approach to stability and invariance properties of ODEs
No full textWe consider nonlinear stability and conservation properties of ODEs with a unifying approach. We show that our theory can be applied to dynamical systems in both the continuous and the discrete case. For this reason, we introduce the concept of pxp sesquilinear forms and pxp-inner products on finite dimensional vector spaces. In particular, we study two classes of functions related to the operators dealing with dissipative, unitary and symplectic ODEs. With this approach, the well-known conditions for BN-stable, unitary and symplectic Runge-Kutta methods are proved in a unifying way
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