A Superlinearly Convergent Scheme for Multi-term Time-fractional Nonlinear Diffusion Equations with Initial Singularity

Multi-term time fractional diffusion model is not only an important physical subject, but also a practical problem commonly involved in engineering. In this paper, we apply the alternating segment technique to combine the classical explicit and implicit schemes, and propose a parallel nature difference method alternating segment pure explicit–implicit (PASE-I) and alternating segment pure implicit–explicit (PASI-E) difference schemes for multi-term time fractional order diffusion equations. The existence and uniqueness of the solutions are proved, and stability and convergence analysis of the two schemes are also given. Theoretical analyses and numerical experiments show that the PASE-I and PASI-E schemes are unconditionally stable and satisfy second-order accuracy in spatial precision and 2 − α order in time precision. When the computational accuracy is equivalent, the CPU time of the two schemes are reduced by up to 2 / 3 compared with the classical implicit difference method. It indicates that the PASE-I and PASI-E parallel difference methods are efficient and feasible for solving multi-term time fractional diffusion equations.

In this paper, we construct a linearized finite difference scheme for a class of multi‐term time‐fractional nonlinear diffusion equations (MTFNDEs) with initial boundary value conditions. Due to the initial singularity of the solutions of MTFNDEs, it is well known that many existing numerical methods for MTFNDEs often suffer from the phenomenon of order reduction. In order to solve the above problem, based on the technique of variable transformation , we propose an improved Euler method to solve MTFNDEs. Then, we prove that the scheme has order accuracy in time for . Meanwhile, the unconditional stability and convergence of the scheme are proved through the energy method. Finally, numerical experiments have been conducted to support the theoretical results and efficiency of our proposed scheme.

A novel multi-term time-fractional mixed diffusion and diffusion-wave equation will be considered in this work. Different from the general multi-term time-fractional mixed diffusion and diffusion-wave equations, this new multi-term equation possesses a special time-fractional operator on the spatial derivative. We use a new discrete scheme to approximate the time-fractional derivative, which can improve the temporal accuracy. Then, a fully discrete spectral scheme is developed based on finite difference discretization in time and Legendre spectral approximation in space. Meanwhile, a very important lemma is proposed and proved, to obtain the unconditional stability and convergence of the fully discrete spectral scheme. Finally, four numerical experiments are presented to confirm our theoretical analysis. Both of our analysis and numerical test indicate that the fully discrete scheme is accurate and efficient in solving the generalized multi-term time-fractional mixed diffusion and diffusion-wave equation.

In this paper, a fully discrete scheme for amulti-term time fractional diffusion equation with initialsingularity is considered. First, we construct a semi-discretescheme by using the $L1$ scheme on a graded mesh for the multi-termCaputo-type time fractional derivatives. It is shownthat with appropriate choice of the grading parameter $r$, themethod has the optimal $2-\\alpha_1$ order convergence, where$\\alpha_1~\\in~(0,1)$ is the highest fractional derivative order inthe multi-term time fractional derivatives. Then we design a fullydiscrete scheme by combining the spectral method for the spatialdiscretization. Convergence and unconditional stability are provedrigorously. In order to reduce the storage requirement andcomputational cost, a fast evaluation scheme for the time fractionalderivatives is applied. Numerical tests confirm that our erroranalysis is sharp and the fast algorithm improves the computationalefficiency significantly.

This work focuses on the design and analysis of a numerical scheme based on a high-order compact ADI difference scheme for a 2D multi-term time-fractional convection diffusion equation subjected to weakly singular solution. In the proposed technique, each time-fractional derivative (TFD) is approximated by means of the L1 scheme on graded mesh and the spatial derivatives are approximated by means of an alternating-direction implicit (ADI) difference scheme of fourth-order on uniform mesh. Stability and convergence analysis of the method are carefully studied. We propose a framework to investigate both H 1 -norm and L 2 -norm optimal error estimates of the suggested scheme. Numerical experiment is carried out to demonstrate the performance of the present numerical scheme and to validate the theoretical error estimates.

IIn this paper, two finite difference methods are used to solve the one-dimensional second order wave equation with constant coefficients subject to specified initial and boundary conditions. Two numerical experiments are considered. The two methods are Central in Time and Central in Space scheme with second order accuracy in both time and space, abbreviated as CTCS (2,2) and Central in Time and Central in Space scheme with second order accuracy in time and fourth order accuracy in space, abbreviated as CTCS (2,4). Properties such as consistency and stability are studied. We also perform spectral analysis of dispersive and dissipative properties of the two methods. Two numerical experiments are considered, and the numerical results are displayed.

L1 Fourier spectral methods for a class of generalized two-dimensional time fractional nonlinear anomalous diffusion equations

Crank–Nicolson extrapolation and finite element method for the Oldroyd fluid with the midpoint rule

In this paper we are going to derive two numerical methods for solving the coupled nonlinear Schrodinger-Boussinesq equation. The first method is a nonlinear implicit scheme of second order accuracy in both directions time and space; the scheme is unconditionally stable. The second scheme is a nonlinear implicit scheme of second order accuracy in time and fourth order accuracy in space direction. A generalized method is also derived where the previous schemes can be obtained by some special values of l. The proposed methods will produced a coupled nonlinear tridiagonal system which can be solved by fixed point method. The exact solutions and the conserved quantities for two different tests are used to display the robustness of the proposed schemes.

In this paper, numerical theory based on the mixed finite-element method and finite difference analog of the Caputo fractional derivative for multi-term time-fractional diffusion equations and diffusion-wave equations is analyzed. The unconditional stability and convergence results are proved for the two resulting fully discrete schemes. Finally, the obtained results are supported by numerical experiments carried out for some test problems.

A new analysis for the convergence of the gradient discretization method for multidimensional time fractional diffusion and diffusion-wave equations

A new approach for increasing the space and time accuracy of numerical schemes employed in standard (commercial) reservoir simulators is presented in this paper. We introduce a novel TVD fully second order flux approximation (Two Time Level Flux TTLF) extending the work previously done on applying TVD schemes [1] to general purpose Black Oil/EOS Compositional reservoir simulators (eg. VIP). The TVD TTLF scheme is a more fundamental application of flux limiters on the second order Taylor's series expansion of the inter-block flux. This enables commercial first order accurate simulators to achieve second order accuracy in both space and time without significantly increasing their complexity. This paper presents the development of the improved temporal accuracy TVD scheme and the results of the scheme on linear and non-linear 1-D pulse propagation problems and 2-D finely gridded compositional problems.

This paper deals with numerical solution to the multi-term time fractional diffusion equation in a finite domain. An implicit finite difference scheme is established based on Caputo's definition to the fractional derivatives, and the upper and lower bounds to the spectral radius of the coefficient matrix of the difference scheme are estimated, with which the unconditional stability and convergence are proved. The numerical results demonstrate the effectiveness of the theoretical analysis, and the method and technique can also be applied to other kinds of time/space fractional diffusion equations.

In this paper, a unified lattice Boltzmann model is developed to solve the multiterm time-fractional mixed diffusion and diffusion-wave equations. To obtain the standard form of the equation, the Caputo derivative terms are approximated using numerical differentiation and composite integration rules. By selecting appropriate auxiliary distribution functions and equilibrium distribution functions, the macroscopic equation is properly recovered. A series of numerical simulations are performed, demonstrating that the results of the proposed method agree well with the analytical solutions. The efficiency of the proposed lattice Boltzmann model is thus validated.

Anisotropic linear triangle finite element approximation for multi-term time-fractional mixed diffusion and diffusion-wave equations with variable coefficient on 2D bounded domain