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R Du - One of the best experts on this subject based on the ideXlab platform.

  • a compact difference scheme for the fractional diffusion wave equation
    Applied Mathematical Modelling, 2010
    Co-Authors: R Du
    Abstract:

    Abstract This article is devoted to the study of high Order difference methods for the fractional diffusion-wave equation. The time fractional derivatives are described in the Caputo’s sense. A compact difference scheme is presented and analyzed. It is shown that the difference scheme is unconditionally convergent and stable in L ∞ -norm. The Convergence Order is O ( τ 3 - α + h 4 ) . Two numerical examples are also given to demonstrate the theoretical results.

Mostafa Abbaszadeh - One of the best experts on this subject based on the ideXlab platform.

Antoine Tambue - One of the best experts on this subject based on the ideXlab platform.

  • optimal strong Convergence rates of numerical methods for semilinear parabolic spde driven by gaussian noise and poisson random measure
    arXiv: Numerical Analysis, 2019
    Co-Authors: Jean Daniel Mukam, Antoine Tambue
    Abstract:

    This paper deals with the numerical approximation of semilinear parabolic stochastic partial differential equation (SPDE) driven simultaneously by Gaussian noise and Poisson random measure, more realistic in modeling real world phenomena. The SPDE is discretized in space with the standard finite element method and in time with the linear implicit Euler method or an exponential integrator, more efficient and stable for stiff problems. We prove the strong Convergence of the fully discrete schemes toward the mild solution. The results reveal how Convergence Orders depend on the regularity of the noise and the initial data.In addition, we exceed the classical Orders $1/2$ in time and $1$ in space achieved in the literature when dealing with SPDE driven by Poisson measure with less regularity assumptions on the nonlinear drift function. In particular, for trace class multiplicative Gaussian noise we achieve Convergence Order $\mathcal{O}(h^2+\Delta t^{1/2})$.For additive trace class Gaussian noise and an appropriate jump function, we achieve Convergence Order $\mathcal{O}(h^2+\Delta t)$. Numerical experiments to sustain the theoretical results are provided.

  • optimal strong Convergence rates of numerical methods for semilinear parabolic spde driven by gaussian noise and poisson random measure
    Computers & Mathematics With Applications, 2019
    Co-Authors: Jean Daniel Mukam, Antoine Tambue
    Abstract:

    Abstract This paper deals with the numerical approximation of semilinear parabolic stochastic partial differential equation (SPDE) driven simultaneously by Gaussian noise and Poisson random measure, more realistic in modeling real world phenomena. The SPDE is discretized in space with the standard finite element method and in time with the linear implicit Euler method or an exponential integrator, more efficient and stable for stiff problems. We prove the strong Convergence of the fully discrete schemes toward the mild solution. The results reveal how Convergence Orders depend on the regularity of the noise and the initial data. In addition, we exceed the classical Orders 1 ∕ 2 in time and 1 in space achieved in the literature when dealing with SPDE driven by Poisson measure with less regularity assumptions on the nonlinear drift function. In particular, for trace class multiplicative Gaussian noise we achieve Convergence Order O ( h 2 + Δ t 1 ∕ 2 ) . For additive trace class Gaussian noise and an appropriate jump function, we achieve Convergence Order O ( h 2 + Δ t ) . Numerical experiments to sustain the theoretical results are provided.

  • strong Convergence of the linear implicit euler method for the finite element discretization of semilinear spdes driven by multiplicative or additive noise
    Applied Mathematics and Computation, 2019
    Co-Authors: Antoine Tambue, Jean Daniel Mukam
    Abstract:

    Abstract This paper aims to investigate the numerical approximation of a general second Order parabolic stochastic partial differential equation(SPDE) driven by multiplicative or additive noise. The linear operator is not necessary self-adjoint, so more useful in concrete applications. The SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. The corresponding scheme is more stable and efficient to solve stochastic advection-dominated reactive transport in porous media. This extends the current results in the literature to not necessary self-adjoint operator. As a challenge, key part of the proof does not rely anymore on the spectral decomposition of the linear operator. The results reveal how the Convergence Orders depend on the regularity of the noise and the initial data. In particular for multiplicative trace class noise, we achieve optimal Convergence Order O ( h 2 + Δ t 1 / 2 ) and for additive trace class noise, we achieve optimal Convergence Order in space and sup-optimal Convergence Order in time of the form O ( h 2 + Δ t 1 − ϵ ) , for an arbitrarily small ϵ > 0. Numerical experiments to sustain our theoretical results are provided.

Man Luo - One of the best experts on this subject based on the ideXlab platform.

Zhizhong Sun - One of the best experts on this subject based on the ideXlab platform.

  • numerical algorithm with high spatial accuracy for the fractional diffusion wave equation with neumann boundary conditions
    Journal of Scientific Computing, 2013
    Co-Authors: Jincheng Ren, Zhizhong Sun
    Abstract:

    A fourth-Order compact algorithm is discussed for solving the time fractional diffusion-wave equation with Neumann boundary conditions. The $$L1$$ discretization is applied for the time-fractional derivative and the compact difference approach for the spatial discretization. The unconditional stability and the global Convergence of the compact difference scheme are proved rigorously, where a new inner product is introduced for the theoretical analysis. The Convergence Order is $$\mathcal{O }(\tau ^{3-\alpha }+h^4)$$ in the maximum norm, where $$\tau $$ is the temporal grid size and $$h$$ is the spatial grid size, respectively. In addition, a Crank---Nicolson scheme is presented and the corresponding error estimates are also established. Meanwhile, a compact ADI difference scheme for solving two-dimensional case is derived and the global Convergence Order of $$\mathcal{O }(\tau ^{3-\alpha }+h_1^4+h_2^4)$$ is given. Then extension to the case with Robin boundary conditions is also discussed. Finally, several numerical experiments are included to support the theoretical results, and some comparisons with the Crank---Nicolson scheme are presented to show the effectiveness of the compact scheme.

  • a box type scheme for fractional sub diffusion equation with neumann boundary conditions
    Journal of Computational Physics, 2011
    Co-Authors: Xuan Zhao, Zhizhong Sun
    Abstract:

    Combining Order reduction approach and L1 discretization, a box-type scheme is presented for solving a class of fractional sub-diffusion equation with Neumann boundary conditions. A new inner product and corresponding norm with a Sobolev embedding inequality are introduced. A novel technique is applied in the proof of both stability and Convergence. The global Convergence Order in maximum norm is O(@t^2^-^@a+h^2). The accuracy and efficiency of the scheme are checked by two numerical tests.