scholarly journals Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 485 ◽  
Author(s):  
Eyaya Fekadie Anley ◽  
Zhoushun Zheng
Keyword(s):  
Finite Difference ◽  
Variable Coefficients ◽  
Convective Transport ◽  
Transport Processes ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Convection Diffusion ◽  
Integer Order ◽  
Difference Formula ◽  
The Right

Space non-integer order convection–diffusion descriptions are generalized form of integer order convection–diffusion problems expressing super diffusive and convective transport processes. In this article, we propose finite difference approximation for space fractional convection–diffusion model having space variable coefficients on the given bounded domain over time and space. It is shown that the Crank–Nicolson difference scheme based on the right shifted Grünwald–Letnikov difference formula is unconditionally stable and it is also of second order consistency both in temporal and spatial terms with extrapolation to the limit approach. Numerical experiments are tested to verify the efficiency of our theoretical analysis and confirm order of convergence.

h2- and h4-extrapolation in eigenvalue problems

1964 ◽  
Vol 60 (1) ◽  
pp. 67-74 ◽  
Author(s):  
S. C. R. Dennis
Keyword(s):  
Finite Difference ◽  
Eigenvalue Problems ◽  
Correction Term ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
General Level ◽  
End Points ◽  
Difference Formula ◽  
End Conditions ◽  
Image Position

Two recent papers have discussed eigenvalue problems relating to second-order, self-adjoint differential equations from the point of view of the deferred approach to the limit in the finite-difference treatment of the problem. In both cases the problem is made definite by considering the differential equationprimes denoting differentiation with respect to x, with two-point boundary conditionsand given at the ends of the interval (0, 1). The usual finite-difference approach is to divide the range (0, 1) into N equal strips of length h = 1/N, giving a set of N + 1 pivotal values φn as the analogue of a solution of (1), φn denoting the pivotal value at x = nh. In terms of central differences we then haveand retaining only second differences yields a finite-difference approximation φn = Un to (1), where the pivotal U-values satisfy the equationsdefined at all internal points, together with two equations holding at the end-points and approximately satisfying the end conditions (2). Here Λ is the corresponding approximation to the eigenvalue λ. A possible finite-difference treatment of the end conditions (2) would be to replace (1) at x = 0 by the central-difference formulaand use the corresponding result for the first derivative of φ, i.e.whereq(x) = λρ(x) – σ(x). Eliminating the external value φ–1 between these two and making use of (1) and (2) we obtain the equationwhere for convenience we write k0 = B0/A0. Similarly at x = 1 we obtainwithkN = B1/A1. If we neglect terms in h3 in these two they become what are usually taken to be the first approximation to the end conditions (2) to be used in conjunction with the set (4) (with the appropriate change φ = U, λ = Λ). This, however, results in a loss of accuracy at the end-points over the general level of accuracy of the set (4), which is O(h4), so there is some justification for retaining the terms in h3, e.g. if a difference correction method were being used they would subsequently be added as a correction term.

scholarly journals Mathematical Modelling and Simulation of Hydrotropic Delignification

2019 ◽  
Vol 13 (1) ◽  
pp. 31 ◽  
Author(s):  
Indah Hartati ◽  
Wahyudi Budi Sediawan ◽  
Hary Sulistyo ◽  
Muhammad Mufti Azis ◽  
Moh Fahrurrozi
Keyword(s):  
Finite Difference ◽  
Quantitative Description ◽  
Transport Processes ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Soluble Product ◽  
Promising Alternative ◽  
Proposed Model ◽  
Hydrotropic Agent ◽  
Hydrotropic Delignification

A B S T R A C TDelignification is a fundamental step in bio-refinery for lignocellulose feedstock processing. Hydrotropic delignification is considered as a promising alternative compared to other conventional delignification processes due to the use of mild chemicals. In this paper, a quantitative description of hydrotropic delignification for a cylindrical biomass particle is presented by using fundamental concepts of chemical kinetics and transport processes. The development of hydrotropic delignification model was based on following assumptions: i) lignin in the biomass is immobile, ii) delignification is considered as a simultaneous process which involves intra-particle diffusion of hydrotropic agent followed by second order reaction for lignin and hydrotropic chemical, as well as intra-particle product diffusion. Finite difference approximation was applied to solve the resulting partial and ordinary differential equations. The simulation results of the proposed model may describe the concentration profiles of lignin, hydrotropic agent and soluble product distributions in a cylindrical solid particle as a function of radial position and time. In addition, the model could also predict the concentration of hydrotropic agent and soluble product in the liquid phase as well as the yield and conversion as a function of time. A local sensitivity analysis method using one factor at a time (OFAT), has been applied to investigate the influence of particle size and hydrotropic agent concentration to the yield and conversion of the hydrotropic delignification model. Validation of the proposed model was conducted by comparing the numerical results with an analytical solution for a simple case diffusion in cylinder with constant surface concentration and in the absence of chemical reaction. The validation result showed that the hydrotropic delignification model was in good agreement with the analytical solution.Keywords: cylindrical particle; delignification; hydrotropic; modelling; simulation A B S T R A KDelignifikasi merupakan tahap penting dalam proses biorefineri biomassa berlignoselulosa. Delignifikasi hidrotropi adalah salah satu alternative proses yang memiliki beberapa kelebihan dibandingkan proses-proses delignifikasi konvensional karena tidak menggunakan bahan kimia berbahaya. Dalam artikel ini disajikan deskripsi kuantitatif proses delignifikasi hidrotropi untuk partikel berbentuk silinder dengan menggunakan konsep fundamental kinetika reaksi dan proses-proses perpindahan. Penyusunan model proses delignifikasi hidrotropi dilakukan berdasarkan asumsi-asumsi bahwa i) lignin pada biomassa bersifat immobile, ii) proses delignifikasi dipandang sebagai suatu rangkaian proses simultan yang terdiri atas proses difusi intrapartikel senyawa hidrotrop, reaksi order dua terhadap lignin dan senyawa hidrotrop, serta difusi intrapartikel produk delignifikasi. Finite difference approximation (FDA) digunakan untuk menyelesaikan persamaan simultan berbentuk persamaan diferensial ordiner dan persamaan diferensial parsial dalam tahap pemodelan. Hasil simulasi memberikan gambaran profil distribusi konsentrasi lignin, konsentrasi senyawa hidrotrop dan produk delignifikasi di dalam partikel padatan yang berbentuk silinder sebagai fungsi posisi dan waktu. Model yang dikembangkan juga dapat memprediksi konsentrasi senyawa hidrotropik dan produk di fasa cairan, serta yield dan konversi sebagai fungsi waktu.  Metode analisis sensitivitas lokal, yakni metode one factor at a time (OFAT), digunakan untuk mengkaji pengaruh ukuran partikel dan konsentrasi senyawa hidrotropik terhadap yield dan konversi proses delignifikasi. Validasi model yang diajukan dilakukan dengan membandingkan hasil analisa numerik dengan hasil penyelesaian analitis untuk kasus difusi pada silinder dengan konsentrasi permukaan yang konstan serta tidak melibatkan reaksi kimia. Hasil validasi model menunjukkan bahwa model delignifikasi hidrotropi yang diajukan memiliki kesesuaian yang tinggi dengan hasil penyelesaian analitis.Kata kunci: delignifikasi; hidrotropi; pemodelan; silinder; simulasi

Wavelet-optimized compact finite difference method for convection–diffusion equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mani Mehra ◽  
Kuldip Singh Patel ◽  
Ankita Shukla
Keyword(s):  
Finite Difference ◽  
Two Dimensions ◽  
Finite Difference Approximation ◽  
Diffusion Equations ◽  
Difference Approximation ◽  
Uniform Grid ◽  
Adaptive Grids ◽  
Smooth Functions ◽  
Compact Finite Difference ◽  
Convection Diffusion

AbstractIn this article, compact finite difference approximations for first and second derivatives on the non-uniform grid are discussed. The construction of diffusion wavelets using compact finite difference approximation is presented. Adaptive grids are obtained for non-smooth functions in one and two dimensions using diffusion wavelets. High-order accurate wavelet-optimized compact finite difference (WOCFD) method is developed to solve convection–diffusion equations in one and two dimensions on an adaptive grid. As an application in option pricing, the solution of Black–Scholes partial differential equation (PDE) for pricing barrier options is obtained using the proposed WOCFD method. Numerical illustrations are presented to explain the nature of adaptive grids for each case.

scholarly journals Numerical Computation and Stability Analysis for the Fractional Subdiffusions with Spatial Variable Coefficients

2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Lei Ren
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Fourth Order ◽  
Difference Method ◽  
Variable Coefficients ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Formula ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method

In this paper, we propose an efficient compact finite difference method for a class of time-fractional subdiffusion equations with spatially variable coefficients. Based on the L2-1σ approximation formula of the time-fractional derivative and a fourth-order compact finite difference approximation to the spatial derivative, an efficient compact finite difference method is developed. The local truncation error and the solvability of the developed method are discussed in detail. The unconditional stability of the resulting scheme and also its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Numerical examples are provided to demonstrate the accuracy and the theoretical results.

scholarly journals Finite difference approximation of a parabolic problem with variable coefficients

2014 ◽  
Vol 95 (109) ◽  
pp. 49-62 ◽  
Author(s):  
Bosko Jovanovic ◽  
Zorica Milovanovic
Keyword(s):  
Finite Difference ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Variable Coefficients ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Convergence Rate Estimate

We study the convergence of a finite difference scheme that approximates the third initial-boundary-value problem for a parabolic equation with variable coefficients on a unit square. We assume that the generalized solution of the problem belongs to the Sobolev space W s,s/2 2, s?3. An almost second-order convergence rate estimate (with additional logarithmic factor) in the discrete W 1,1/2 2 norm is obtained. The result is based on some nonstandard a priori estimates involving fractional order discrete Sobolev norms.

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