Repeat Exercise 3-132 using the implicit method. Give the...

Name: Problem 141, Chapter 3: Basic Heat And Mass Transfer
Uploaded: 2020-08-19T03:08:22-08:00
Duration: 5 min 17 s
Channel: Ryan Conley
Description: Repeat Exercise 3-132 using the implicit method. Give the finite-difference approximations for the given mesh, and use Gauss-Seidel iteration to perform three iterations at the first time step. Also check the temperatures at the first time step using a direct method.

Repeat Exercise 3-132 using the implicit method. Give the finite-difference approximations for the given mesh, and use Gauss-Seidel iteration to perform three iterations at the first time step. Also check the temperatures at the first time step using a direct method.

Key Concepts

Direct Methods for Solving Linear Systems

Direct methods, such as Gaussian elimination or matrix factorization techniques, involve solving the system of algebraic equations in a finite number of operations. These methods are used to check the accuracy of the iterative solution by computing an exact (or nearly exact) solution in a single computational step, which can serve as a benchmark for the iterative methods.

Gauss-Seidel Iteration

Gauss-Seidel iteration is an iterative technique used to solve systems of linear equations that arise from discretized differential equations. It works by sequentially updating each variable using the latest available values, thereby accelerating the convergence towards the solution of the entire system. This method is especially useful when the system matrix is sparse and has favorable properties such as diagonal dominance.

Implicit Finite Difference Method

The implicit finite difference method approximates partial differential equations by discretizing both time and spatial derivatives, with the time derivative being evaluated at a future time level. This method often leads to a system of algebraic equations that must be solved simultaneously and is known for its unconditional stability under certain conditions, making it suitable for stiff problems or when larger time steps are desired.

Finite-Difference Approximations

Finite-difference approximations replace the derivatives in a differential equation with difference quotients based on values at discrete mesh points. This involves selecting appropriate schemes (such as backward, forward, or central differences) for time and space, converting the continuous problem into a system of algebraic equations that approximate the original differential equation over a given mesh.

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