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Magazine Chapter 10: Problem 3
Show that the forward-time central space scheme for the one-way wave equation is accurate of order \([1,2]\).
Short Answer
Expert verified
The FTCS scheme is accurate of order \([1,2]\) for time and space.
Step by step solution
01
Understanding the Forward-Time Central-Space Scheme
The forward-time central-space (FTCS) scheme is a numerical method used to approximate solutions of differential equations, specifically ones like the one-way wave equation: \( u_t + cu_x = 0 \). This method involves discretizing time and space into small steps, where time is incremented forward and space is centered.
02
Set up the FTCS Scheme
Given the one-way wave equation \( u_t + cu_x = 0 \), the FTCS scheme can be expressed as:\[\frac{u_{i}^{n+1} - u_{i}^{n}}{\Delta t} + c \frac{u_{i+1}^{n} - u_{i-1}^{n}}{2\Delta x} = 0\]This equation uses forward differences in time and central differences in space.
03
Expand Using Taylor Series
To determine the order of accuracy, expand \( u_{i}^{n+1} \), \( u_{i+1}^{n} \), and \( u_{i-1}^{n} \) using Taylor series expansions around \( (x_i, t_n) \):- \( u_{i}^{n+1} = u(t_n + \Delta t, x_i) \approx u(t_n, x_i) + \Delta t u_t + \frac{(\Delta t)^2}{2} u_{tt} + \cdots \)- \( u_{i+1}^{n} \approx u(t_n, x_i) + \Delta x u_x + \frac{(\Delta x)^2}{2} u_{xx} + \cdots \)- \( u_{i-1}^{n} \approx u(t_n, x_i) - \Delta x u_x + \frac{(\Delta x)^2}{2} u_{xx} + \cdots \)
04
Substitute Taylor Expansions into FTCS Scheme
Substitute the Taylor expansions from step 3 into the FTCS formula:\[\left( u(t_n, x_i) + \Delta t u_t + \cdots \right) - u(t_n, x_i) \]\[ + c \left( \frac{u(t_n, x_i) + \Delta x u_x + \cdots - u(t_n, x_i) + \Delta x u_x + \cdots}{2 \Delta x} \right) = 0\]This simplifies to:\[u_t + cu_x + \frac{(\Delta x)^2}{3!} u_{xxx} = 0 \]
05
Determine the Order of Accuracy
The error terms from the Taylor expansions determine the accuracy. From the expansion, we note that the errors are of order \( \mathcal{O}(\Delta t) \) for time and \( \mathcal{O}(\Delta x^2) \) for space. Therefore, the FTCS scheme is first-order accurate in time and second-order accurate in space.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Finite Difference Schemes
The concept of finite difference schemes is a cornerstone of numerical methods for solving partial differential equations (PDEs). These schemes involve approximating derivatives by using calculated differences between function values at specific points.
- The forward-time central-space (FTCS) scheme is a specific type of finite difference scheme.
- It involves discretizing the time variable with a forward difference and the space variable with a central difference.
- This method is specifically used for differential equations such as the one-way wave equation.
Taylor Series Expansion
The Taylor series expansion is a method used in numerical analysis to approximate functions through series of polynomial expressions.
- For any given function, it expands the function into an infinite sum of terms calculated from the values of its derivatives at a specific point.
- The main idea in the FTCS scheme is to use Taylor series expansions to approximate the values of function derivatives.
- This allows us to understand how the function behaves in both time and space increments.
One-way Wave Equation
The one-way wave equation is a fundamental problem that describes how waves, like sound or light waves, propagate in a given direction. It is often expressed in the form \( u_t + cu_x = 0 \), where:
- \( u_t \) represents the time derivative, indicating how the wave amplitude changes over time.
- \( u_x \) represents the spatial derivative, showing how the wave amplitude changes in space.
- \( c \) is the wave speed, a constant dictating how fast the wave travels.
Order of Accuracy
In numerical methods, the order of accuracy is a measure of how closely an approximated solution converges to the exact solution as the mesh size approaches zero.
- The order represents the rate at which the error decreases as the step sizes (\( \Delta t \) and \( \Delta x \)) become smaller.
- In the FTCS scheme, the time component is accurate to first order \( \mathcal{O}(\Delta t) \).
- The space component is accurate to second order \( \mathcal{O}(\Delta x^2) \).
