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Show that the FTBS scheme of the uni-directional wave equation is con- vergent. First explain what it means to be convergent and then state any theorem that you use and show that the hypothesis is satisfied.

Show that the FTBS scheme of the uni-directional wave equation is con- vergent. First explain what it means to be convergent and then state any theorem that you use and show that the hypothesis is satisfied.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 10: Convergence of the FTBS Scheme for the Uni-Directional Wave Equation** **Objective:** To demonstrate that the FTBS (Forward-Time Backward-Space) scheme for the uni-directional wave equation is convergent. This involves: 1. Explaining the concept of convergence. 2. Stating a relevant theorem to use. 3. Showing that the hypotheses of the theorem are satisfied. **Step-by-Step Solution:** 1. **Concept of Convergence:** - Convergence in the context of numerical schemes typically refers to the property that as the grid spacing (both in time and space) is refined, the numerical solution approaches the exact solution of the differential equation. Mathematically, this means that the error between the numerical and exact solutions tends to zero as the grid resolution increases. 2. **Relevant Theorem:** - One popular theorem used to demonstrate the convergence of numerical schemes is the **Lax Equivalence Theorem**. The theorem states: "A consistent finite difference approximation for a well-posed linear initial-value problem is convergent if and only if it is stable." - To apply this theorem, we need to examine both the consistency and the stability of the FTBS scheme. 3. **Consistency:** - A numerical scheme is consistent if the truncation error (the error made by the difference between the partial differential equation and its finite difference approximation) tends to zero as the grid spacings tend to zero. - For the FTBS scheme applied to the one-dimensional wave equation, write down the finite difference approximation and verify that the truncation error decreases with grid refinement. 4. **Stability:** - Stability typically involves showing that the errors do not grow uncontrollably as the computation proceeds in time. A common method to prove stability is through von Neumann stability analysis. - Perform a von Neumann stability analysis on the FTBS scheme to ensure that the error terms do not amplify with time steps. 5. **Conclusion:** - If both consistency and stability are demonstrated for the FTBS scheme, apply the Lax Equivalence Theorem to conclude that the scheme is convergent. By following these steps, one can show the convergence of the FTBS scheme for the uni-directional wave equation, establishing that the numerical solutions will approximate the exact solutions closely as the computational grid is refined. --- This problem involves detailed mathematical steps and may include graphs or diagrams related to truncation error analysis
Transcribed Image Text:**Problem 10: Convergence of the FTBS Scheme for the Uni-Directional Wave Equation** **Objective:** To demonstrate that the FTBS (Forward-Time Backward-Space) scheme for the uni-directional wave equation is convergent. This involves: 1. Explaining the concept of convergence. 2. Stating a relevant theorem to use. 3. Showing that the hypotheses of the theorem are satisfied. **Step-by-Step Solution:** 1. **Concept of Convergence:** - Convergence in the context of numerical schemes typically refers to the property that as the grid spacing (both in time and space) is refined, the numerical solution approaches the exact solution of the differential equation. Mathematically, this means that the error between the numerical and exact solutions tends to zero as the grid resolution increases. 2. **Relevant Theorem:** - One popular theorem used to demonstrate the convergence of numerical schemes is the **Lax Equivalence Theorem**. The theorem states: "A consistent finite difference approximation for a well-posed linear initial-value problem is convergent if and only if it is stable." - To apply this theorem, we need to examine both the consistency and the stability of the FTBS scheme. 3. **Consistency:** - A numerical scheme is consistent if the truncation error (the error made by the difference between the partial differential equation and its finite difference approximation) tends to zero as the grid spacings tend to zero. - For the FTBS scheme applied to the one-dimensional wave equation, write down the finite difference approximation and verify that the truncation error decreases with grid refinement. 4. **Stability:** - Stability typically involves showing that the errors do not grow uncontrollably as the computation proceeds in time. A common method to prove stability is through von Neumann stability analysis. - Perform a von Neumann stability analysis on the FTBS scheme to ensure that the error terms do not amplify with time steps. 5. **Conclusion:** - If both consistency and stability are demonstrated for the FTBS scheme, apply the Lax Equivalence Theorem to conclude that the scheme is convergent. By following these steps, one can show the convergence of the FTBS scheme for the uni-directional wave equation, establishing that the numerical solutions will approximate the exact solutions closely as the computational grid is refined. --- This problem involves detailed mathematical steps and may include graphs or diagrams related to truncation error analysis
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