The following is a brief summary of the methodology and results of my Masters summer internship at NIT Kurukshetra.
The numerical solution of nonlinear partial differential equations (PDEs) is crucial in various scientific and engineering fields. In this study, we implemented Newton’s Method, Quasi-Linearization Method, and the Finite Difference Method (FDM) to solve different types of nonlinear PDEs, including the Burgers’ equation. The Newton-Raphson scheme was utilized to convert nonlinear PDEs into a system of linear equations, which were then solved using iterative techniques. Similarly, the Quasi-Linearization Method was applied to linearize nonlinear PDEs for efficient computation. The implementation was carried out using Python, leveraging numerical techniques to ensure accuracy, stability, and computational efficiency. The results demonstrate the effectiveness of these methods in handling nonlinear PDEs, providing valuable insights into their numerical behavior and practical applications.
The nonlinear Partial Differential Equation (PDE) was solved using the Finite Difference Method (FDM), Quasi-Linearization Method, and Newton-Raphson iterative technique. The procedure converts the nonlinear PDE into a system of algebraic equations that can be solved efficiently.
Consider a general nonlinear PDE of the form:
For this project, Burgers' equation was used as a benchmark problem:
\[ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2} \]where \(u\) represents the dependent variable and \(\nu\) denotes the viscosity coefficient.
The spatial and temporal derivatives were approximated using finite difference schemes.
\[ \frac{\partial u}{\partial t} \approx \frac{u_i^{n+1}-u_i^n}{\Delta t} \] \[ \frac{\partial u}{\partial x} \approx \frac{u_{i+1}^n-u_{i-1}^n}{2\Delta x} \] \[ \frac{\partial^2u}{\partial x^2} \approx \frac{u_{i+1}^n-2u_i^n+u_{i-1}^n} {\Delta x^2} \]Substituting these approximations into the PDE yields a nonlinear algebraic system:
\[ F(U)=0 \]The nonlinear system was linearized around the current approximation \(U^{(k)}\) using Taylor series expansion:
\[ F(U^{(k+1)}) \approx F(U^{(k)}) + J(U^{(k)}) \left( U^{(k+1)}-U^{(k)} \right) \]where \(J(U^{(k)})\) is the Jacobian matrix of the nonlinear system.
The correction vector \(\Delta U^{(k)}\) was obtained by solving:
\[ J(U^{(k)})\Delta U^{(k)} = -F(U^{(k)}) \]The solution was then updated as:
\[ U^{(k+1)} = U^{(k)} + \Delta U^{(k)} \]Iterations were continued until the correction norm satisfied:
\[ ||\Delta U^{(k)}|| < 10^{-6} \]or until the residual norm became sufficiently small:
\[ ||F(U^{(k)})|| < 10^{-6} \]The developed numerical framework is applicable to a wide range of nonlinear PDE problems arising in fluid dynamics, heat transfer, engineering simulations, and financial modeling.
Consider the following system of nonlinear equations:
$$ f_1(x,y) = xy + x^2 − 3 = 0 $$ $$ f_(x,y) = y^2 − 5x − 9 = 0 $$Define the nonlinear vector function:
$$ F(x)= [f_1(x,y),f_2(x,y)]^T = [ xy+x^2−3, y^2−5x−9]^TThe Jacobian matrix is given by:
\[ J(x)= \begin{bmatrix} \dfrac{\partial f_1}{\partial x} & \dfrac{\partial f_1}{\partial y} \\ \dfrac{\partial f_2}{\partial x} & \dfrac{\partial f_2}{\partial y} \end{bmatrix} = \begin{bmatrix} y+2x & x \\ -5 & 2y \end{bmatrix} \]The Newton-Raphson iteration formula is:
\[ J\!\left(X^{(k)}\right)\Delta X^{(k)} = - F\!\left(X^{(k)}\right) \] \[ X^{(k+1)} = X^{(k)} + \Delta X^{(k)} \]Using Python implementation, the approximate solution obtained is: $$x = 0.7064556018364156, y = 3.5400957627172622 $$
Consider the partial differential equation:
\[ \frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial x^2} =0 \]Subject to the boundary and initial conditions:
\[ u(0,t)=0 \] \[ u(2,t)=1 \] \[ u(x,0)=-0.2\sin\left(\frac{\pi x}{L}\right) \]The PDE was discretized using the Finite Difference Method (FDM) and solved numerically for different mesh sizes.
| \(N=M\) | \(v\) | \(w\) |
|---|---|---|
| 10 | 0.053817 | 1.236107 |
| 20 | 0.025887 | 1.184467 |
| 30 | 0.017080 | 1.165735 |
| 40 | 0.0126606 | 1.1561760 |
Consider the nonlinear PDE:
\[ u_t = u\,u_x \]with boundary conditions:
\[ u(0,t)=2,\qquad u(5,t)=4,\qquad T=0.1,\qquad L=5 \]The nonlinear term was linearized using the Quasi-Linearization method and solved numerically on a mesh with:
\[ N_x = 5,\qquad N_t = 5 \]| \(x_0\) | \(x_1\) | \(x_2\) | \(x_3\) | \(x_4\) | \(x_5\) | |
|---|---|---|---|---|---|---|
| \(t_0\) | 2 | 1 | 1 | 1 | 1 | 4 |
| \(t_1\) | 2 | 1.52020202 | 1 | 1 | -0.515 | 4 |
| \(t_2\) | 2 | 2.29893428 | 1.26930497 | 1.765075 | 0.26930497 | 4 |
| \(t_3\) | 2 | 3.13786316 | 1.62827342 | 2.64333302 | -0.03516866 | 4 |
| \(t_4\) | 2 | 3.68497884 | 2.06610441 | 4.81495916 | -0.0113788 | 4 |
| \(t_5\) | 2 | 1 | 1 | 1 | 1 | 4 |