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We propose a discontinuous Galerkin method for fractional convection-diffusion equations with a superdiffusion operator of order α(1 < α < 2) defined through the fractional Laplacian. The fractional operator of order α is expressed as a composite of first order derivatives and a fractional integral of order 2 – α. The fractional convection-diffusion problem is expressed as a system of low order differential/integral equations, and a local discontinuous Galerkin method scheme is proposed for the equations. We prove stability and optimal order of convergence 𝓞(hk+1) for the fractional diffusion problem, and an order of convergence of 𝓞(hk+½) is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.

In this work, we have used reduced differential transform method (RDTM) to compute an approximate solution of the Two-Dimensional Convection-Diffusion equations (TDCDE). This method provides the solution quickly in the form of a convergent series. Also, by using RDTM the approximate solution of two-dimensional convection-diffusion equation is obtained. Further, we have computed exact solution of non-homogeneous CDE by using the same method. To the best of my knowledge, the research work carried out in the present paper has not been done, and is new. Examples are provided to support our work.

In this paper we intend to establish fast numerical approaches to solve a class of initial-boundary problem of time-space fractional convection–diffusion equations. We present a new unconditionally stable implicit difference method, which is derived from the weighted and shifted Grünwald formula, and converges with the second-order accuracy in both time and space variables. Then, we show that the discretizations lead to Toeplitz-like systems of linear equations that can be efficiently solved by Krylov subspace solvers with suitable circulant preconditioners. Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from O ( N 2 ) to O ( N ) and the computational complexity from O ( N 3 ) to O ( N log N ) in each iterative step, where N is the number of grid nodes. Extensive numerical examples are reported to support our theoretical findings and show the utility of these methods over traditional direct solvers of the implicit difference method, in terms of computational cost and memory requirements.

Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations (PDEs) because they are flexible with respect to geometry, they can provide high order convergence, they allow for local refinement, and they are easy to implement in higher dimensions. For global RBF methods, one of the major disadvantages is the computational cost associated with the dense linear systems that arise. Therefore, research is currently directed towards localized RBF approximations such as the RBF partition of unity collocation method (RBF–PUM) proposed here. The objective of this paper is to establish that RBF–PUM is viable for parabolic PDEs of convection–diffusion type. The stability and accuracy of RBF–PUM is investigated partly theoretically and partly numerically. Numerical experiments show that high-order algebraic convergence can be achieved for convection–diffusion problems. Numerical comparisons with finite difference and pseudospectral methods have been performed, showing that RBF–PUM is competitive with respect to accuracy, and in some cases also with respect to computational time. As an application, RBF–PUM is employed for a two-dimensional American option pricing problem. It is shown that using a node layout that captures the solution features improves the accuracy significantly compared with a uniform node distribution.

The paper presents the theory of the space-time Petrov-discontinuous Galerkin finite element (PDGFE) method for the discretization of the nonstationary linear convection-diffusion problems. The PDGFE method is modified for the discontinuous Galerkin finite element (DGFE) method in the case of the symmetric interior penalty Galerkin (SIPG) scheme. PDGFE method is applied separately in space using different space gride on different time levels. We prove the properties of the bilinear form a PD, m ( u, ν ) ( V − elliptic and continuity) stability and prove the approximate solution converges with the error of order o ( h 2 + τ 3 ). A numerical experiment is carried out to confirm the theoretical conclusions.

In this article, we consider the time-dependent convection-diffusion-reaction equation coupled with the Darcy equation. We propose a numerical scheme based on finite element methods for the discretization in space and the implicit Euler method for the discretization in time. We establish optimal a posteriori error estimates with two types of computable error indicators, the first one linked to the time discretization and the second one to the space discretization. Finally, numerical investigations are performed and presented.

In this paper, based on the previous work (Shi and Guo in Phys Rev E 79:016701, 2009 ), we develop a multiple-relaxation-time (MRT) lattice Boltzmann model for general nonlinear anisotropic convection–diffusion equation (NACDE), and show that the NACDE can be recovered correctly from the present model through the Chapman–Enskog analysis. We then test the MRT model through some classic CDEs, and find that the numerical results are in good agreement with analytical solutions or some available results. Besides, the numerical results also show that similar to the single-relaxation-time lattice Boltzmann model or so-called BGK model, the present MRT model also has a second-order convergence rate in space. Finally, we also perform a comparative study on the accuracy and stability of the MRT model and BGK model by using two examples. In terms of the accuracy, both the analysis and numerical results show that a numerical slip on the boundary would be caused in the BGK model, and cannot be eliminated unless the relaxation parameter is fixed to be a special value, while the numerical slip in the MRT model can be overcome once the relaxation parameters satisfy some constrains. The results in terms of stability also demonstrate that the MRT model could be more stable than the BGK model through tuning the free relaxation parameters.

In this paper, an efficient lattice Boltzmann model for n -dimensional steady convection–diffusion equation with variable coefficients is proposed through modifying the equilibrium distribution function properly, and the Chapman–Enskog analysis shows that the steady convection–diffusion equation with variable coefficients can be recovered exactly. Detailed simulations are performed to test the model, and the results show that the accuracy and efficiency of the present model are better than previous models.

In this paper, a novel numerical model for the Black–Scholes equations is developed. To address some potential issues that may arise when solving this equation using the conventional model, the original Black–Scholes equation is reformulated as a convection–diffusion equation. The Crank–Nicolson scheme is utilized to discretize the diffusion and source terms in time, and the convection term is dealt explicitly using the Adams–Bashforth method. Spatial derivatives for the diffusion term are approximated with the central difference scheme. However, the spatial discretization of the convection term can be chosen adaptively according to the problem setting and the parameter values. By employing various discretization schemes for the convection term, a series of models are constructed for solving the Black–Scholes equations. We demonstrated that the discrete matrix with the new models has a better diagonal dominance property than that of the conventional model, which is beneficial for the solution. Subsequently, the new model is validated with several single‐asset and multiasset benchmark problems, and the numerical results are compared with analytic solutions. It is observed that the choice of the discretization scheme for the convection term has little impact on the computational efficiency but can be significant on the accuracy and robustness. We find that the new model shows clear advantages over conventional methods, particularly in stable market conditions.

The space fractional advection–diffusion equation is a crucial type of fractional partial differential equation, widely used for its ability to more accurately describe natural phenomena. Due to the complexity of analytical approaches, this paper focuses on its numerical investigation. A lattice Boltzmann model for the spatial fractional convection–diffusion equation is developed, and an error analysis is carried out. The spatial fractional convection–diffusion equation is solved for several examples. The validity of the model is confirmed by comparing its numerical solutions with those obtained from other methods The results demonstrate that the lattice Boltzmann method is an effective tool for solving the space fractional convection–diffusion equation.