Communications on Applied Mathematics and Computation

A Note on Numerical Algorithm for the Time-Caputo and Space-Riesz Fractional Difusion Equation

Junhong Tian, Hengfei Ding

2021, 3(4): 571-584. doi:10.1007/s42967-021-00139-0

摘要 ( 6219 )

PDF

参考文献 | 相关文章 | 多维度评价

Recently, Zhang and Ding developed a novel fnite diference scheme for the timeCaputo and space-Riesz fractional difusion equation with the convergence order $\mathcal{O}\left(\tau^{2-\alpha}+h^{2}\right)$ in Zhang and Ding (Commun. Appl. Math. Comput. 2(1): 57–72, 2020). Unfortunately, they only gave the stability and convergence results for α ∈ (0, 1) and $\beta \in\left[\frac{7}{8}+\frac{\sqrt[3]{621+48 \sqrt{87}}}{24}+\frac{19}{8 \sqrt[3]{621+48 \sqrt{87}}}, 2\right]$. In this paper, using a new analysis method, we fnd that the original diference scheme is unconditionally stable and convergent with order $\mathcal{O}\left(\tau^{2-\alpha}+h^{2}\right)$ for all α ∈ (0, 1) and β ∈ (1, 2]. Finally, some numerical examples are given to verify the correctness of the results.

Existence of Boundary Value Problems for Impulsive Fractional Diferential Equations with a Parameter

Jin You, Mengrui Xu, Shurong Sun

2021, 3(4): 585-604. doi:10.1007/s42967-021-00145-2

摘要 ( 2238 )

PDF

参考文献 | 相关文章 | 多维度评价

We investigate a class of boundary value problems for nonlinear impulsive fractional differential equations with a parameter. By the deduction of Altman’s theorem and Krasnoselskii’s fxed point theorem, the existence of this problem is proved. Examples are given to illustrate the efectiveness of our results.

Preface to Focused Section on Efcient HighcOrder Time Discretization Methods for Partial Diferential Equations

Sebastiano Boscarino, Giuseppe Izzo, Lorenzo Pareschi, Giovanni Russo, Chi-Wang Shu

2021, 3(4): 605-605. doi:10.1007/s42967-021-00164-z

摘要 ( 6848 )

PDF

F. L. Romeo, M. Dumbser, M. Tavelli

2021, 3(4): 607-647. doi:10.1007/s42967-020-00077-3

摘要 ( 3856 )

PDF

参考文献 | 相关文章 | 多维度评价

A new high-order accurate staggered semi-implicit space-time discontinuous Galerkin (DG) method is presented for the simulation of viscous incompressible fows on unstructured triangular grids in two space dimensions. The staggered DG scheme defnes the discrete pressure on the primal triangular mesh, while the discrete velocity is defned on a staggered edge-based dual quadrilateral mesh. In this paper, a new pair of equal-order-interpolation velocity-pressure fnite elements is proposed. On the primary triangular mesh (the pressure elements), the basis functions are piecewise polynomials of degree N and are allowed to jump on the boundaries of each triangle. On the dual mesh instead (the velocity elements), the basis functions consist in the union of piecewise polynomials of degree N on the two subtriangles that compose each quadrilateral and are allowed to jump only on the dual element boundaries, while they are continuous inside. In other words, the basis functions on the dual mesh are built by continuous fnite elements on the subtriangles. This choice allows the construction of an efcient, quadraturefree and memory saving algorithm. In our coupled space-time pressure correction formulation for the incompressible Navier-Stokes equations, the arbitrary high order of accuracy in time is achieved through the use of time-dependent test and basis functions, in combination with simple and efcient Picard iterations. Several numerical tests on classical benchmarks confrm that the proposed method outperforms existing staggered semi-implicit space-time DG schemes, not only from a computer memory point of view, but also concerning the computational time.

Parallel Implicit-Explicit General Linear Methods

Steven Roberts, Arash Sarshar, Adrian Sandu

2021, 3(4): 649-669. doi:10.1007/s42967-020-00083-5

摘要 ( 2013 )

PDF

参考文献 | 相关文章 | 多维度评价

High-order discretizations of partial diferential equations (PDEs) necessitate high-order time integration schemes capable of handling both stif and nonstif operators in an efcient manner. Implicit-explicit (IMEX) integration based on general linear methods (GLMs) ofers an attractive solution due to their high stage and method order, as well as excellent stability properties. The IMEX characteristic allows stif terms to be treated implicitly and nonstif terms to be efciently integrated explicitly. This work develops two systematic approaches for the development of IMEX GLMs of arbitrary order with stages that can be solved in parallel. The frst approach is based on diagonally implicit multi-stage integration methods (DIMSIMs) of types 3 and 4. The second is a parallel generalization of IMEX Euler and has the interesting feature that the linear stability is independent of the order of accuracy. Numerical experiments confrm the theoretical rates of convergence and reveal that the new schemes are more efcient than serial IMEX GLMs and IMEX Runge–Kutta methods.

Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity

Zheng Sun, Chi-Wang Shu

2021, 3(4): 671-700. doi:10.1007/s42967-020-00098-y

摘要 ( 2114 )

PDF

参考文献 | 相关文章 | 多维度评价

A time discretization method is called strongly stable (or monotone), if the norm of its numerical solution is nonincreasing. Although this property is desirable in various of contexts, many explicit Runge-Kutta (RK) methods may fail to preserve it. In this paper, we enforce strong stability by modifying the method with superviscosity, which is a numerical technique commonly used in spectral methods. Our main focus is on strong stability under the inner-product norm for linear problems with possibly non-normal operators. We propose two approaches for stabilization: the modifed method and the fltering method. The modifed method is achieved by modifying the semi-negative operator with a high order superviscosity term; the fltering method is to post-process the solution by solving a difusive or dispersive problem with small superviscosity. For linear problems, most explicit RK methods can be stabilized with either approach without accuracy degeneration. Furthermore, we prove a sharp bound (up to an equal sign) on difusive superviscosity for ensuring strong stability. For nonlinear problems, a fltering method is investigated. Numerical examples with linear non-normal ordinary diferential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance.

High Order Semi-implicit Multistep Methods for Time-Dependent Partial Diferential Equations

Giacomo Albi, Lorenzo Pareschi

2021, 3(4): 701-718. doi:10.1007/s42967-020-00110-5

摘要 ( 3406 )

PDF

参考文献 | 相关文章 | 多维度评价

We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great fexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-difusion equation and in the setting of strong stability preserving (SSP) methods. Our fndings are demonstrated on several examples, including nonlinear reaction-difusion and convection-difusion problems.

Strong Stability Preserving IMEX Methods for Partitioned Systems of Diferential Equations

Giuseppe Izzo, Zdzisław Jackiewicz

2021, 3(4): 719-758. doi:10.1007/s42967-021-00158-x

摘要 ( 2103 )

PDF

参考文献 | 相关文章 | 多维度评价

We investigate strong stability preserving (SSP) implicit-explicit (IMEX) methods for partitioned systems of diferential equations with stif and nonstif subsystems. Conditions for order p and stage order q = p are derived, and characterization of SSP IMEX methods is provided following the recent work by Spijker. Stability properties of these methods with respect to the decoupled linear system with a complex parameter, and a coupled linear system with real parameters are also investigated. Examples of methods up to the order p = 4 and stage order q = p are provided. Numerical examples on six partitioned test systems confrm that the derived methods achieve the expected order of convergence for large range of stepsizes of integration, and they are also suitable for preserving the accuracy in the stif limit or preserving the positivity of the numerical solution for large stepsizes.

当期目录