Two high-order time discretization schemes for subdiffusion problems with nonsmooth data

2020 ◽  
Vol 23 (5) ◽  
pp. 1349-1380
Author(s):  
Yanyong Wang ◽  
Yubin Yan ◽  
Yan Yang
Keyword(s):  
Time Discretization ◽  
Fixed Time ◽  
High Order ◽  
Transform Method ◽  
Order Of Accuracy ◽  
Nonsmooth Data ◽  
High Order Schemes ◽  
Discretization Schemes ◽  
High Order Time ◽  
Convergence Orders

Abstract Two new high-order time discretization schemes for solving subdiffusion problems with nonsmooth data are developed based on the corrections of the existing time discretization schemes in literature. Without the corrections, the schemes have only a first order of accuracy for both smooth and nonsmooth data. After correcting some starting steps and some weights of the schemes, the optimal convergence orders O(k 3–α ) and O(k 4–α ) with 0 < α < 1 can be restored for any fixed time t for both smooth and nonsmooth data, respectively. The error estimates for these two new high-order schemes are proved by using Laplace transform method for both homogeneous and inhomogeneous problem. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

scholarly journals Positivity preserving high order schemes for angiogenesis models

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Carpio ◽  
E. Cebrian
Keyword(s):  
Kinetic Equation ◽  
Time Discretization ◽  
Strong Stability ◽  
High Order ◽  
Angiogenic Factor ◽  
Strong Stability Preserving ◽  
Positivity Preserving ◽  
Blood Vessel Formation ◽  
High Order Schemes ◽  
Asymptotic Reduction

Abstract Hypoxy induced angiogenesis processes can be described by coupling an integrodifferential kinetic equation of Fokker–Planck type with a diffusion equation for the angiogenic factor. We propose high order positivity preserving schemes to approximate the marginal tip density by combining an asymptotic reduction with weighted essentially non oscillatory and strong stability preserving time discretization. We capture soliton-like solutions representing blood vessel formation and spread towards hypoxic regions.

High-Order Schemes Combining the Modified Equation Approach and Discontinuous Galerkin Approximations for the Wave Equation

2012 ◽  
Vol 11 (2) ◽  
pp. 691-708 ◽  
Author(s):  
Cyril Agut ◽  
Julien Diaz ◽  
Abdelaâziz Ezziani
Keyword(s):  
Wave Equation ◽  
Discontinuous Galerkin ◽  
Time Discretization ◽  
High Order ◽  
High Order Method ◽  
High Order Schemes ◽  
Galerkin Approximations ◽  
Space Discretization ◽  
New Interpretation ◽  
Modified Equation

AbstractWe present a new high order method in space and time for solving the wave equation, based on a new interpretation of the “Modified Equation” technique. Indeed, contrary to most of the works, we consider the time discretization before the space discretization. After the time discretization, an additional biharmonic operator appears, which can not be discretized by classical finite elements. We propose a new Discontinuous Galerkin method for the discretization of this operator, and we provide numerical experiments proving that the new method is more accurate than the classical Modified Equation technique with a lower computational burden.

scholarly journals Fourth Order Difference Approximations for Space Riemann-Liouville Derivatives Based on Weighted and Shifted Lubich Difference Operators

2014 ◽  
Vol 16 (2) ◽  
pp. 516-540 ◽  
Author(s):  
Minghua Chen ◽  
Weihua Deng
Keyword(s):  
Fractional Derivatives ◽  
Computational Cost ◽  
Variable Coefficients ◽  
High Order ◽  
Linear Multistep Methods ◽  
Difference Operators ◽  
Fractional Operators ◽  
High Order Schemes ◽  
Discretization Schemes ◽  
Low Order

AbstractHigh order discretization schemes play more important role in fractional operators than classical ones. This is because usually for classical derivatives the stencil for high order discretization schemes is wider than low order ones; but for fractional operators the stencils for high order schemes and low order ones are the same. Then using high order schemes to solve fractional equations leads to almost the same computational cost with first order schemes but the accuracy is greatly improved. Using the fractional linear multistep methods, Lubich obtains thev-th order (v <6) approximations of theα-th derivative (α >0) or integral (α <0) [Lubich, SIAM J. Math. Anal., 17, 704-719, 1986], because of the stability issue the obtained scheme can not be directly applied to the space fractional operator withαЄ (1,2) for time dependent problem. By weighting and shifting Lubich’s 2nd order discretization scheme, in [Chen & Deng, SINUM, arXiv:1304.7425] we derive a series of effective high order discretizations for space fractional derivative, called WSLD operators there. As the sequel of the previous work, we further provide new high order schemes for space fractional derivatives by weighting and shifting Lubich’s 3rd and 4th order discretizations. In particular, we prove that the obtained 4th order approximations are effective for space fractional derivatives. And the corresponding schemes are used to solve the space fractional diffusion equation with variable coefficients.

scholarly journals Toward error estimates for general space-time discretizations of the advection equation

2020 ◽  
Vol 23 (1-4) ◽  
Author(s):  
Martin J. Gander ◽  
Thibaut Lunet
Keyword(s):  
Error Estimates ◽  
Time Discretization ◽  
Space Time ◽  
Error Tolerance ◽  
Advection Equation ◽  
Order Of Accuracy ◽  
One Dimensional ◽  
General Space ◽  
Discretization Schemes ◽  
The One

AbstractWe develop new error estimates for the one-dimensional advection equation, considering general space-time discretization schemes based on Runge–Kutta methods and finite difference discretizations. We then derive conditions on the number of points per wavelength for a given error tolerance from these new estimates. Our analysis also shows the existence of synergistic space-time discretization methods that permit to gain one order of accuracy at a given CFL number. Our new error estimates can be used to analyze the choice of space-time discretizations considered when testing Parallel-in-Time methods.

scholarly journals Error Estimates of a Continuous Galerkin Time Stepping Method for Subdiffusion Problem

2021 ◽  
Vol 88 (3) ◽  
Author(s):  
Yuyuan Yan ◽  
Bernard A. Egwu ◽  
Zongqi Liang ◽  
Yubin Yan
Keyword(s):  
Piecewise Linear ◽  
Time Discretization ◽  
Convergence Order ◽  
Step Size ◽  
Time Step ◽  
Transform Method ◽  
Nonsmooth Data ◽  
Time Step Size ◽  
Time Stepping ◽  
Continuous Galerkin

AbstractA continuous Galerkin time stepping method is introduced and analyzed for subdiffusion problem in an abstract setting. The approximate solution will be sought as a continuous piecewise linear function in time t and the test space is based on the discontinuous piecewise constant functions. We prove that the proposed time stepping method has the convergence order $$O(\tau ^{1+ \alpha }), \, \alpha \in (0, 1)$$ O ( τ 1 + α ) , α ∈ ( 0 , 1 ) for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where $$\tau $$ τ is the time step size. This convergence order is higher than the convergence orders of the popular convolution quadrature methods (e.g., Lubich’s convolution methods) and L-type methods (e.g., L1 method), which have only $$O(\tau )$$ O ( τ ) convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity.

Strong Stability-Preserving High-Order Time Discretization Methods

SIAM Review ◽  
2001 ◽  
Vol 43 (1) ◽  
pp. 89-112 ◽  
Author(s):  
Sigal Gottlieb ◽  
Chi-Wang Shu ◽  
Eitan Tadmor
Keyword(s):  
Time Discretization ◽  
Strong Stability ◽  
High Order ◽  
Strong Stability Preserving ◽  
Discretization Methods ◽  
High Order Time
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