ILM FAN YANGILIKLARI KONFERENSIYASI
IYUL
ANDIJON,2025
160
NUMERICAL SOLUTIONS OF NONLINEAR DIFFUSION PROBLEMS WITH
CONVECTIVE TRANSFER
Gulom Urolov
Institution: National University of Uzbekistan
Email:
gulomurolov95@gmail.com
Introduction
Nonlinear diffusion problems coupled with convective (advection) transport arise in many
physical and engineering systems such as heat and mass transfer, environmental flows, and
reactive transport in porous media. These problems are modeled using partial differential
equations (PDEs) that include nonlinear diffusion terms and advection terms, making analytical
solutions difficult or impossible. Therefore, robust and accurate numerical methods are
necessary to analyze such systems.
Mathematical Formulation
The general form of a nonlinear diffusion-convection equation is given by:
∂u/∂t =
∇
·(D(u)
∇
u) - v·
∇
u + S(x, t, u)
Here:
- u(x,t) is the scalar field (e.g., temperature, concentration),
- D(u) is a nonlinear diffusion coefficient,
- v is the velocity vector representing convection,
- S(x,t,u) is a source or sink term.
Numerical Methods
To numerically solve the above PDE, we use the Finite Difference Method (FDM) for spatial
discretization. Depending on the problem, either explicit or implicit time integration schemes
may be applied. For nonlinear systems, implicit methods such as the Backward Euler or Crank-
Nicolson scheme are preferred due to their stability.
For a one-dimensional case, the equation simplifies to:
∂u/∂t = ∂/∂x [ D(u) ∂u/∂x ] - v ∂u/∂x
Using finite difference approximation:
(u_i^{n+1} - u_i^n)/Δt = (1/Δx)[ D_{i+1/2}(u_{i+1}^n - u_i^n)/Δx - D_{i-1/2}(u_i^n -
u_{i-1}^n)/Δx ] - v (u_i^n - u_{i-1}^n)/Δx
The resulting system of equations is nonlinear due to D(u), and iterative solvers like Newton-
Raphson or Picard iteration are used to compute the solution at each time step.
Implementation
The numerical schemes are implemented in MATLAB and Python. The user defines the initial
and boundary conditions, diffusion coefficient D(u), velocity field v, and the source term. The
numerical solver is designed to be flexible and adaptable to various real-world problems.
Validation and Results
Benchmark problems with known analytical or reference solutions are used to validate the
accuracy of the numerical methods. The convergence, stability, and computational performance
ILM FAN YANGILIKLARI KONFERENSIYASI
IYUL
ANDIJON,2025
161
of the methods are analyzed. Visualization of results includes contour plots, surface plots, and
error analysis to demonstrate the correctness of the approach.
Conclusion
Nonlinear diffusion problems with convective transfer are complex but highly relevant to many
scientific and engineering fields. Numerical methods provide a powerful tool for simulating
and analyzing such systems. The finite difference approach combined with implicit time
integration and iterative solvers results in accurate and stable solutions for a variety of
nonlinear problems.
References:
1. 1. J. Crank, The Mathematics of Diffusion, 2nd Edition, Oxford University Press, 1975.
2. 2. R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations,
SIAM, 2007.
3. 3. C. Hirsch, Numerical Computation of Internal and External Flows, Vol. 1, John Wiley &
Sons, 1988.
4. 4. G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference
Methods, Oxford University Press, 1985.
5. 5. Randall J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992.
6. 6. W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-Dependent Advection-
Diffusion-Reaction Equations, Springer, 2003.
