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Magazine Chapter 13: Problem 18
Solve the following equation numerically using the Crank-Nicolson procedure. $$ \frac{\partial^{2} f(x, t)}{\partial x^{2}}=\frac{\partial f(x, t)}{\partial t} $$ for \(0 \leq x \leq 1\) with a step length \(h=0.2\) and \(0 \leq t \leq 0.6\) with a step length \(k=0.04\) where \(f(x, 0)=x^{2}+x-1\) and \(f(0, t)=2 t-1, f(1, t)=1+2 t\)
Short Answer
Expert verified
Use Crank-Nicolson method to iteratively solve the linear system at each time step with boundary and initial conditions.
Step by step solution
01
Define the Crank-Nicolson Scheme
The Crank-Nicolson method is a finite difference method used for the numerical solution of partial differential equations. It is an implicit method that averages the results of the Forward Time Centered Space (FTCS) and Backward Time Centered Space (BTCS) methods. For the PDE given, the Crank-Nicolson scheme is:ewlineewline\[ \frac{f_{i}^{n+1} - f_{i}^{n}}{k} = \frac{1}{2} \left( \frac{f_{i+1}^{n} - 2f_{i}^{n} + f_{i-1}^{n}}{h^2} + \frac{f_{i+1}^{n+1} - 2f_{i}^{n+1} + f_{i-1}^{n+1}}{h^2} \right) \]
02
Initial and Boundary Conditions
Given:\(f(x, 0) = x^2 + x - 1\)ewline\(f(0, t) = 2t - 1\)ewline\(f(1, t) = 1 + 2t\)Calculate the initial values at each spatial point using the initial condition \(f(x, 0) = x^2 + x - 1\). For \(h = 0.2\) and \(0 \leq x \leq 1\) with \(N = 5\) (i.e., \(0, 0.2, 0.4, 0.6, 0.8, 1.0\)):ewline\[f_{i}^{0} = f(x_{i}, 0) = x_{i}^{2} + x_{i} - 1\]
03
Discretize the Spatial and Temporal Intervals
Divide the spatial domain from 0 to 1 into 5 equal segments (\(h = 0.2\)) resulting in points: \(x = [0, 0.2, 0.4, 0.6, 0.8, 1.0] \). The time domain from 0 to 0.6 with step length \(k = 0.04\) results in time increments: \( t = [0, 0.04, 0.08, ...., 0.6] \).
04
Set up the Equation System
At each time step, set up a system of linear equations for the interior points (\( i = 1, 2, 3, 4 \)). The system of equations is given by:ewline\( - \frac{k}{2h^2} f_{i-1}^{n+1} + (1 + \frac{k}{h^2}) f_{i}^{n+1} - \frac{k}{2h^2} f_{i+1}^{n+1} = \frac{k}{2h^2} f_{i-1}^{n} + (1 - \frac{k}{h^2}) f_{i}^{n} + \frac{k}{2h^2} f_{i+1}^{n} \)ewlineFor boundary conditions:\( f_{0}^{n+1} = 2t - 1 \) for the left boundary.\( f_{5}^{n+1} = 1 + 2t \) for the right boundary.
05
Iterate Over Time Steps
Use the initial state \(f_{i}^{0}\) and the boundary conditions to solve the system of equations at each time step iteratively. Update \(f_{i}^{n+1}\) at each spatial point using the Crank-Nicolson scheme derived in Step 4.
06
Calculate Using Matrix Formulation
Using the matrix formulation, solve for \( f^{n+1} \) for each time step, where:ewline\( A \cdot f^{n+1} = B \cdot f^{n} + C \)ewlineHere, matrices A and B incorporate the coefficients from the Crank-Nicolson method and vector C incorporates the boundary conditions.
07
Short Answer
Compute the specific solution values at each point by solving the linear systems for each time step iteratively.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Numerical Solution of PDEs
Partial Differential Equations (PDEs) model various physical phenomena like heat conduction and wave propagation. Solving PDEs analytically is often challenging due to their complexity. The numerical solutions provide approximate answers using computational methods. One such method is the Crank-Nicolson procedure. This approach is crucial in science and engineering to simulate real-world systems.
Finite Difference Method
The finite difference method approximates derivatives by using differences between function values at discrete points. For example, it approximates the derivative \(f'(x)\) using the difference between \(f(x+h)\) and \(f(x)\) divided by \(h\). In the Crank-Nicolson method, both forward and backward differences are averaged, making it more accurate and stable.
Initial and Boundary Conditions
Initial and boundary conditions are essential for solving PDEs numerically. They define the values of the function at the start and boundaries of the domain. For our problem:
- The initial condition is \(f(x,0) = x^2 + x - 1\).
- The boundary conditions are \(f(0,t) = 2t - 1\) and \(f(1,t) = 1 + 2t\).
Discretization
Discretization converts continuous variables into discrete points. For our spatial domain \(0 \leq x \leq 1\) and a step length \(h = 0.2\), we have points \(\text{x}=[0, 0.2, 0.4, 0.6, 0.8, 1.0]\). For the time domain \(0 \leq t \leq 0.6\) with step length \(k = 0.04\), we have points \(t=[0, 0.04, 0.08, 0.12,..., 0.6]\). Discretizing helps in setting up a grid for applying numerical methods.
Linear Equations System
After discretizing, we formulate a system of linear equations for each time step. For inner points \(i=1,2,3,4\), the system derived is:
\[ - \frac{\text{k}}{2h^2} f_{i-1}^{n+1} + (1 + \frac{\text{k}}{h^2}) f_{i}^{n+1} - \frac{\text{k}}{2h^2} f_{i+1}^{n+1} = \frac{\text{k}}{2h^2} f_{i-1}^{n} + (1 - \frac{\text{k}}{h^2}) f_{i}^{n} + \frac{\text{k}}{2h^2} f_{i+1}^{n} \]
This system needs to be solved iteratively at each time step.
\[ - \frac{\text{k}}{2h^2} f_{i-1}^{n+1} + (1 + \frac{\text{k}}{h^2}) f_{i}^{n+1} - \frac{\text{k}}{2h^2} f_{i+1}^{n+1} = \frac{\text{k}}{2h^2} f_{i-1}^{n} + (1 - \frac{\text{k}}{h^2}) f_{i}^{n} + \frac{\text{k}}{2h^2} f_{i+1}^{n} \]
This system needs to be solved iteratively at each time step.
Matrix Formulation
Using matrix notation simplifies solving large systems of linear equations. We denote the system as:
\[A \cdot f^{n+1} = B \cdot f^{n} + C\]
Here, matrices \(A\) and \(B\) represent coefficients from the discretized Crank-Nicolson scheme, and \(C\) includes boundary conditions. This formulation allows efficient use of computational techniques for solving the equations.
\[A \cdot f^{n+1} = B \cdot f^{n} + C\]
Here, matrices \(A\) and \(B\) represent coefficients from the discretized Crank-Nicolson scheme, and \(C\) includes boundary conditions. This formulation allows efficient use of computational techniques for solving the equations.
Iterative Solution
To solve the system iteratively, start from initial conditions and boundary constraints, then compute the values at each step. Use:
- The Crank-Nicolson scheme derived system to update \(f_{i}^{n+1}\).
- Iterate this process for each time step \(k\).
- Ensure convergence for accuracy.
