Numerical Analysis: What Is It?

Coverprefacecontentsnumeical Analysiswhat Is It1 Mathematical Prelim

Prove that the Runge-Kutta method, k1 = h f(tn, yn), k2 = h f ( tn + 1 2 h, yn + 1 2 k1 ) , k3 = h f ( tn + 3 4 h, yn + 3 4 k2 ) , yn+1 = yn + k1 + 3k2 + 4k3, has local truncation error O ( h4 ).

Consider the linear multistep method yn+1 = p ∑_{j=0}^{a} a_j y_{n−j} + h p ∑_{j=0}^{a} b_j f(t_{n−j}, y_{n−j}) for the problem y′(t) = f(t, y(t)), y(t0) = y0. Assuming that the method uses uniform time steps h and that y ∈ C^{m+1}([t0, tf]), prove that it has local truncation error O ( h^{m+1} ) if and only if p ∑_{j=0}^{a} (−j)^k a_j + k p ∑_{j=0}^{a} (−j)^{k−1} b_j = 1, for k = 0, 1, ..., m. Hint: replace each f with an appropriate y' and Taylor expand around tn.

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The analysis of numerical methods for ordinary differential equations (ODEs) is fundamental for understanding their accuracy and stability in approximating solutions. Two significant classes of methods—the Runge-Kutta methods and linear multistep methods—are widely used in computational mathematics. Here, we examine the local truncation errors associated with these methods, focusing on their order and the conditions necessary for their accuracy.

Runge-Kutta Method and Its Truncation Error

The Runge-Kutta method is acclaimed for its simplicity and high order of accuracy. The specific three-stage Runge-Kutta method presented is characterized by the following iterative formulas:

  • k1 = h f(tn, yn)
  • k2 = h f(tn + 1/2 h, yn + 1/2 k1)
  • k3 = h f(tn + 3/4 h, yn + 3/4 k2)
  • yn+1 = yn + (k1 + 3k2 + 4k3)/6

To determine the local truncation error, one applies a Taylor expansion of the exact solution y(t) around point tn and compares it to the numerical solution. The goal is to show that the residual—the difference between the numerical method's approximation and the Taylor polynomial expansion—is of order h^4, indicating a local truncation error of that order.

Performing the Taylor expansion of y(t) yields:

y(tn + h) = y(tn) + h y'(tn) + (h^2/2) y''(tn) + (h^3/6) y'''(tn) + (h^4/24) y''''(ξ), for some ξ between tn and tn + h.

By matching these derivatives with the derivatives approximated through the Runge-Kutta formulation, it can be demonstrated that the method's local truncation error is proportional to h^4, confirming its order. This involves substituting the derivatives with the method's stages and verifying that the error terms align appropriately, which they do for this Runge-Kutta scheme.

Linear Multistep Methods and Their Truncation Errors

Linear multistep methods extend the idea of single-step methods by incorporating multiple previous points in the approximation. The general form is:

yn+1 = p ∑_{j=0}^{a} a_j y_{n−j} + h p ∑_{j=0}^{a} b_j f(t_{n−j}, y_{n−j}),

where the coefficients a_j and b_j depend on the specific method. These methods are designed to satisfy certain order conditions to ensure a specified degree of accuracy.

The key to analyzing the local truncation error involves expanding the function y(t) and its derivatives into Taylor series around tn. Substituting into the multistep formula and comparing term-by-term with the Taylor expansion of the exact solution reveals the specific conditions on the coefficients necessary for achieving order m+1 accuracy.

The precise conditions for the coefficients, as expressed by the sum conditions, are derived by ensuring that the truncation error terms vanish up to the desired order. This entails confirming that:

p ∑_{j=0}^{a} (−j)^k a_j + k p ∑_{j=0}^{a} (−j)^{k−1} b_j = 1, for all k = 0, 1,..., m.

This set of equations ensures that the linear combinations of previous y-values and function evaluations accurately replicate the derivatives' Taylor series, thus guaranteeing the method's order of accuracy.

Conclusion

Both the Runge-Kutta and linear multistep methods' accuracy hinges on satisfying specific order conditions derived from Taylor series expansions. The Runge-Kutta method's local truncation error of order h^4 reflects its capacity to mimic the Taylor expansion up to the third derivative terms accurately. Simultaneously, the multistep method achieves order m+1 if the coefficients are chosen to satisfy the stated sum conditions. Analyzing and confirming these properties are crucial steps in the development and application of reliable numerical ODE solvers.

References

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