Abstract This paper deals with initial-boundary value problems (IBVPs) for quasilinear parabolic equations with unbounded nonlinearity, mixed derivatives, and reaction coefficients of arbitrary sign. Finite-difference schemes (FDSs) with second-order spatial approximation are constructed for these problems. It is proved that the proposed schemes satisfy a nonstandard maximum principle under separate conditions on the temporal and spatial grid steps, without any coupling restriction between them, thereby ensuring unconditional monotonicity. Moreover, using the Ladyzhenskaya transformation, discrete two-sided estimates for the numerical solution are derived and shown to be fully consistent with the corresponding estimates for the continuous problem. The convergence and stability of the proposed schemes are rigorously justified in the grid L 2 {L_{2}} -norm.
Abstract Richardson extrapolation (RE) is an adaptive numerical technique that computes a linear combination of numerical solutions produced by using the same base method with different time-step sizes in order to improve the accuracy. A qualitative property that we expect from the resulting combined method when stiff problems are to be solved is A-stability, which provides a bounded numerical solution on Dahlquist’s autonomous test problem by any choice of the time-step size. Conditions of A-stability have been extensively studied for certain combinations of Runge–Kutta (RK) methods with several versions of RE. However, the A-stability of a base method is not automatically inherited by the combined method. Moreover, A-stability has some limitations when a non-autonomous initial-value problem is to be solved, e.g., a problem obtained by semi-discretization of a time-dependent PDE. Therefore, a possible generalization of A-stability called AN-stability was advocated. Our aim is to study the conditions under which an RK method combined with RE results in an AN-stable method. Simultaneously, an easily verifiable algebraic condition is gained for A-stability.
Abstract This special issue is dedicated to the memory of Raytcho Lazarov, who passed away on November 30, 2024. The articles collected here reflect some of his principal research interests, particularly in numerical methods for partial differential equations and the development of algorithms for their efficient and reliable application. We briefly highlight his major scientific contributions and provide an overview of the results presented in this volume. The contributions are authored by his former students, colleagues, coauthors, and long-term collaborators – researchers whose work has been deeply influenced by Lazarov’s ideas and achievements. Over more than half a century of remarkably productive scholarship, he made fundamental contributions across a broad range of important areas in computational and applied mathematics.
In this paper, we study a reduced-order model based on proper orthogonal decomposition (POD) for a two-dimensional semilinear parabolic equation. The weak Galerkin finite element method (WG-FEM) is used for spatial discretization and the Crank-Nicolson (CN) scheme is employed for temporal discretization. We introduce the POD formulation and establish the optimal error estimates for both WG-FEM and WG-POD solutions, achieving second-order accuracy in time and O(h(k+1)) accuracy in space in the L-2 norm, and the numerical results are consistent with the theoretical conclusions. This indicates that the method can significantly reduce the degrees of freedom and save CPU time while maintaining high accuracy.
In this paper, we design two classes of numerical schemes for the time-fractional Allen-Cahn equation based on the discrete gradient method and the L1-type formula of the Riemann-Liouville fractional derivative. The suggested schemes are as accurate and efficient as a recent work [H. Liao, T. Tang and T. Zhou, An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen-Cahn equation, SIAM J. Sci. Comput. 43 2021, 5, A3503-A3526] for the time-fractional Allen-Cahn equation with a double-well potential function, but can address the time-fractional Allen-Cahn equation with a general potential function. Thanks to the orthogonal convolution technique on nonuniform time mesh: (i) the first type of schemes unconditionally preserves the discrete energy dissipation law; (ii) the second type of schemes is constructed by introducing the fixed-point iteration and stabilized factor method for the first scheme, and proven to be maximum bound principle preserving. Finally, extensive numerical comparisons are reported to verify the efficiency of the proposed schemes and the correctness of the theoretical results.
