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Algorithms based on DQM with new sets of base functions for solving parabolic partial differential equations in $(2+1)$ dimension

Brajesh Kumar Singh, Pramod Kumar

TL;DR

This work addresses the numerical solution of the time-dependent 2D convection-diffusion equation using three differential quadrature methods based on modified cubic B-spline bases (MTB-DQM, mExp-DQM, mECDQ) for spatial discretization. By transforming the PDE into a first-order ODE system in time, $\frac{dU}{dt} = B U + F$, and advancing with the SSP-RK54 scheme, the authors demonstrate unconditional stability on tested grids through eigenvalue analysis and matrix stability considerations. Numerical experiments with Dirichlet and Neumann boundaries show that the proposed methods achieve high accuracy, often surpassing existing schemes, with cubic convergence for Dirichlet cases (when $\beta_x,\beta_y>0$) and quadratic convergence for Neumann cases. The results indicate that these DQM-based approaches are efficient, high-order, and robust tools for 2D parabolic PDEs in convection-diffusion applications.

Abstract

This paper deals with the numerical computations of two space dimensional time dependent parabolic partial differential equations by adopting adopting an optimal five stage fourth-order strong stability preserving Runge Kutta (SSP-RK54) scheme for time discretization, and three methods of differential quadrature with different sets of modified B-splines as base functions, for space discretization: namely i) mECDQM: (DQM with modified extended cubic B-splines); ii) mExp-DQM: DQM with modified exponential cubic B-splines, and iii) MTB-DQM: DQM with modified trigonometric cubic B-splines. Specially, we implement these methods on convection-diffusion equation to convert them into a system of first order ordinary differential equations,in time which can be solved using any time integration method, while we prefer SSP-RK54 scheme. All the three methods are found stable for two space convection-diffusion equation by employing matrix stability analysis method. The accuracy and validity of the methods are confirmed by three test problems of two dimensional convection-diffusion equation, which shows that the proposed approximate solutions by any of the method are in good agreement with the exact solutions.

Algorithms based on DQM with new sets of base functions for solving parabolic partial differential equations in $(2+1)$ dimension

TL;DR

This work addresses the numerical solution of the time-dependent 2D convection-diffusion equation using three differential quadrature methods based on modified cubic B-spline bases (MTB-DQM, mExp-DQM, mECDQ) for spatial discretization. By transforming the PDE into a first-order ODE system in time, , and advancing with the SSP-RK54 scheme, the authors demonstrate unconditional stability on tested grids through eigenvalue analysis and matrix stability considerations. Numerical experiments with Dirichlet and Neumann boundaries show that the proposed methods achieve high accuracy, often surpassing existing schemes, with cubic convergence for Dirichlet cases (when ) and quadratic convergence for Neumann cases. The results indicate that these DQM-based approaches are efficient, high-order, and robust tools for 2D parabolic PDEs in convection-diffusion applications.

Abstract

This paper deals with the numerical computations of two space dimensional time dependent parabolic partial differential equations by adopting adopting an optimal five stage fourth-order strong stability preserving Runge Kutta (SSP-RK54) scheme for time discretization, and three methods of differential quadrature with different sets of modified B-splines as base functions, for space discretization: namely i) mECDQM: (DQM with modified extended cubic B-splines); ii) mExp-DQM: DQM with modified exponential cubic B-splines, and iii) MTB-DQM: DQM with modified trigonometric cubic B-splines. Specially, we implement these methods on convection-diffusion equation to convert them into a system of first order ordinary differential equations,in time which can be solved using any time integration method, while we prefer SSP-RK54 scheme. All the three methods are found stable for two space convection-diffusion equation by employing matrix stability analysis method. The accuracy and validity of the methods are confirmed by three test problems of two dimensional convection-diffusion equation, which shows that the proposed approximate solutions by any of the method are in good agreement with the exact solutions.

Paper Structure

This paper contains 12 sections, 46 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Description of the methods
  3. The MTB-DQM
  4. The evaluation of the weighting coefficients $a_{ij}^{(r)}$ and $b_{ij}^{(r)} (r=1,2)$
  5. The mExp-DQM
  6. The evaluation of the weighting coefficients $a_{ij}^{(r)}$ and $b_{ij}^{(r)} (r=1,2)$
  7. The mECDQ method
  8. The evaluation of the weighting coefficients $a_{ij}^{(r)}$ and $b_{ij}^{(r)} (r=1,2)$
  9. Implementation of the method for $2$D convection-diffusion equation
  10. Stability of the methods for $2$D convection-diffusion equation
  11. Numerical results and discussions
  12. Conclusions

Figures (2)

  • Figure 1: Eigenvalues $\lambda_1$ and $\lambda_2$ for different grid sizes
  • Figure 5: The approximate (left) and exact solution behavior (right) of $2$D convection-diffusion equation \ref{['eq-CDEs']} with $\alpha_x=\alpha_y=0.10$, $\beta_x=\beta_y=-1$ at $t = 1$