Repeat Exercise 3-133 using the implicit method. Give the...

Name: Problem 143, Chapter 3: Basic Heat And Mass Transfer
Uploaded: 2020-08-19T03:11:32-08:00
Duration: 4 min 30 s
Channel: Ryan Conley
Description: Repeat Exercise 3-133 using the implicit method. Give the finite-difference approximations for the specified mesh, and use Gauss-Seidel iteration to perform five iterations at the first time step. Also check the temperatures at the first time step using a direct method.

Repeat Exercise 3-133 using the implicit method. Give the finite-difference approximations for the specified mesh, and use Gauss-Seidel iteration to perform five iterations at the first time step. Also check the temperatures at the first time step using a direct method.

Key Concepts

Implicit Method

The implicit method is a numerical technique for solving differential equations where the solution at the new time level is defined by equations that include unknown future values. It tends to be more stable than explicit methods, particularly for stiff problems, and typically involves solving a system of equations at each time step.

Finite Difference Approximation

Finite difference approximation involves replacing derivatives in differential equations with discrete difference formulas that estimate the derivative based on the values of the function at specific mesh points. This turning of a differential equation into an algebraic form allows numerical solution over a defined grid.

Mesh Discretization

Mesh discretization refers to dividing the continuous domain into a finite set of discrete points where the equations are approximated. This framework provides the structure needed to apply methods such as finite difference or finite element approaches, and impacts the accuracy and stability of the numerical solution.

Gauss-Seidel Iteration

Gauss-Seidel iteration is an iterative algorithm used to solve systems of linear equations that result from the discretization of differential equations. In this method, successive approximations are refined by using the most recent updates, which can enhance convergence properties especially for systems with favorable structure.

Direct Method

A direct method for solving linear systems uses algorithmic procedures like Gaussian elimination or LU factorization to compute the solution exactly in a finite number of steps, contrasting with iterative methods that converge progressively to a solution. This approach is particularly useful for validating iterative results or solving smaller systems efficiently.

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