Math Problem Statement
Consider the IVP with solution y(t) ( y′(t) = f (t, y), a < t < b y(a) = y0 Using a similar strategy as in the derivation of Explicit Euler, derive Implicit Euler’s method: w0 = y0 wi+1 = wi + hf (ti+1, wi+1) Hint: use Taylor expansion centered at ti+1 and evaluated at ti Note: Implicit Euler is more computationally expensive then Explicit Euler, but it is more stable and gives better results for certain ODEs (stiff ODEs). We will see this later in the class.
Solution
To derive the Implicit Euler method for solving the initial value problem (IVP)
we use a similar approach to the derivation of the Explicit Euler method, but with a key difference in evaluating the derivative. Here's the step-by-step derivation:
Step 1: Taylor Expansion Centered at
The Taylor expansion of about is:
where is some point in .
Rearranging to solve for , and neglecting the higher-order term , we get:
For simplicity, let the step size . This gives:
Step 2: Approximation Using the IVP
From the IVP, . Substituting into the equation, we have:
This is the Implicit Euler method in terms of . Since the exact solution is generally unknown, we approximate it numerically using , the numerical approximation of . Thus, the recurrence relation becomes:
Step 3: Final Implicit Euler Formula
Key Characteristics of Implicit Euler
- Implicit nature: The formula requires solving for at each step, often requiring iterative numerical techniques (e.g., Newton's method).
- Stability: This method is more stable than Explicit Euler, particularly for stiff ODEs, making it suitable for such problems.
Would you like a worked example to illustrate this derivation or more details about its implementation?
5 Follow-Up Questions
- How does the stability of the Implicit Euler method compare to the Explicit Euler method for stiff ODEs?
- What numerical methods can be used to solve for in the Implicit Euler formula?
- Can you derive the Explicit Euler method for comparison with the Implicit Euler method?
- What are the advantages and disadvantages of using Implicit Euler in practice?
- How does the choice of step size affect the accuracy and stability of Implicit Euler?
Tip
When implementing the Implicit Euler method, ensure that your numerical solver for is efficient, as this step is computationally intensive but critical for accurate results.
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Math Problem Analysis
Mathematical Concepts
Ordinary Differential Equations (ODEs)
Numerical Methods
Taylor Series Expansion
Implicit Methods
Formulas
y(t_{i+1}) ≈ y(t_i) + h * y'(t_{i+1})
y'(t) = f(t, y)
w_{i+1} = w_i + h * f(t_{i+1}, w_{i+1})
Theorems
Taylor Series Approximation
Stability of Numerical Methods
Suitable Grade Level
Undergraduate (Numerical Analysis or Differential Equations)
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