数学代写|数值方法作业代写numerical methods代考|CS514

数学代写|数值方法作业代写numerical methods代考|Improved Euler method

The improved Euler method is quite similar to the Euler’s method. In the improved Euler’s method, the approximate solution is improved by using the average of the values at the initially given point and the new point, hence reducing the errors in the Euler’s method. We obtain Eqs. (6.8) $-(6.10)$ as:
$$q_1=f\left(x_n, y_n\right)$$

$$q_2=f\left(x_n+h, y_n+h q_1\right)$$
The average of $q_1$ and $q_2$ is used in the Euler’s method:
$$y_{n+1}=f\left(x_n+h, y_n+h\left(q_1+q_2\right) / 2\right)$$
The calculation of the improved Euler was demonstrated using the problem solved earlier, i.e.,
$$\frac{d y}{d x}=\frac{y \ln y}{2 x}$$
where the initial condition is $y(2)=e$.
From the initial condition, $x_0=2, y_0=e$, the step size of $h=0.1$, where $e=2.7182818$
$$\begin{gathered} q_1=f\left(x_n, y_n\right) \ q_1=f(2, e)=\frac{e \text { Ine }}{2 \times 2}=0.6795704429 \ q_2=f\left(x_n+h, y_n+h q_1\right) \ q_2=f(2+0.1, e+0.1 * 0.6795704429)=f(2.1,2.7862388443) \ q_2=\frac{2.7862388443 \operatorname{In}(2.7862388443)}{2 \times 2.1}=0.6797710311 \ y_{n+1}=f\left(x_n+h, y_n+h\left(q_1+q_2\right) / 2\right) \ y_{n+1}=f\left(2+0.1, e+0.1\left(\frac{0.6795704429+0.6797710311}{2}\right)\right) \ f\left(x_n, y_n\right)=f(2.1,2.7862488737)=\frac{2.7862488737 \operatorname{In}(2.7862488737)}{2 \times 2.1}=0.679775866 \ y_1=2.7182818+0.1 \times 0.679775866=2.7862593866 \end{gathered}$$
From the first term, it is clear that there is a significant difference between the Euler and the improved Euler. Due to the complexity of the improved Euler, the computational solution using the improved Euler is recommended. The computational process will be adequately explained in succeeding sections in the chapter.

数学代写|数值方法作业代写numerical methods代考|RungeeKutta method

Runge-Kutta (RK) methods are referred to as a family of implicit and explicit iterative methods. Like the improved Euler method, RK procedure include the Euler method. There are different types of RK method (explicit, embedded, and implicit). The explicit methods include Forward Euler, Explicit midpoint method, Heun’s method, Ralston’s method, generic second-order method, Kutta’s third-order method, generic third-order method, Heun’s third-order method, Ralston’s third-order method,third-order strong stability preserving Runge-Kutta (SSPRK3), classic fourth-order method, Ralston’s fourth-order method, and 3/8-rule fourth-order method. The embedded methods include Heun-Euler, Fehlberg RK1(2), Bogacki-Shampine, Fehlberg, Cash-Karp, and Dormand-Prince. The implicit methods include the backward Euler, implicit midpoint, Crank-Nicolson method, Gauss-Legendre methods, diagonally implicit Runge-Kutta methods, Lobatto methods, and Radau methods. The most popular RK method is the RK4. It is derived from the initial value problem as the Euler’s method where:
$$\frac{d y}{d x}=f(x, y), \quad y\left(x_0\right)=y_0$$
and the step size is assumed to be greater than zero at all times. The unknown function $y$ is written as:
$$\begin{gathered} y_{n+1}=y_n+\frac{1}{6} h\left(q_1+2 q_2+2 q_3+q_4\right) \ x_{n+1}=x_n+h \end{gathered}$$
For $n>0$,
$$\begin{gathered} q_1=f\left(x_n, y_n\right) \ q_2=f\left(x_n+\frac{h}{2}, y_n+h \frac{q_1}{2}\right) \ q_3=f\left(x_n+\frac{h}{2}, y_n+h \frac{q_2}{2}\right) \ q_4=f\left(x_n+h, y_n+h q_3\right) \end{gathered}$$
The difference between the Euler and RK method is that Euler’s method makes use of the information on the slope or the derivative of $y$ at the given time step to extrapolate the solution to the next time-step. On the other hand, Runge-Kutta methods make use of the information on the “slope” at more than one point to extrapolate the solution to the future time step.

数值方法代考

数学代写|数值方法作业代写numerical methods代考|Improved Euler method

$$\begin{gathered} q_1=f\left(x_n, y_n\right) \ q_2=f\left(x_n+h, y_n+h q_1\right) \end{gathered}$$

$$y_{n+1}=f\left(x_n+h, y_n+h\left(q_1+q_2\right) / 2\right)$$

$$\frac{d y}{d x}=\frac{y \ln y}{2 x}$$

$$q_1=f\left(x_n, y_n\right) q_1=f(2, e)=\frac{e \text { Ine }}{2 \times 2}=0.6795704429 q_2=f\left(x_n+h, y_n+h q_1\right) q_2=f(2+0.1, e+0.1$$

数学代写|数值方法作业代写numerical methods代考|RungeeKutta method

Runge-Kutta (RK) 方法被称为一系列隐式和显式迭代方法。与改进的欧拉方法一样，RK过程也包括欧拉 方法。有不同类型的 RK 方法 (显式、嵌入和隐式) 。显式方法包括前向欧拉法、显式中点法、Heun 法、Ralston法、泛型二阶法、Kutta三阶法、泛型三阶法、Heun三阶法、Ralston三阶法、三阶法强稳定 性保持 Runge-Kutta (SSPRK3)、经典四阶方法、Ralston 四阶方法和 $3 / 8$ 规则四阶方法。嵌入式方法包括 Heun-Euler、Fehlberg RK1(2)、Bogacki-Shampine、Fehlberg、Cash-Karp 和 Dormand-Prince。隐式方 法包括后向欧拉法、隐式中点法、Crank-Nicolson 法、Gauss-Legendre 方法、对角隐式 Runge-Kutta 方 法、Lobatto 方法和 Radau 方法。最流行的 RK方法是 RK4。它源自作为欧拉方法的初值问题，其中:
$$\frac{d y}{d x}=f(x, y), \quad y\left(x_0\right)=y_0$$

$$y_{n+1}=y_n+\frac{1}{6} h\left(q_1+2 q_2+2 q_3+q_4\right) x_{n+1}=x_n+h$$

$$q_1=f\left(x_n, y_n\right) q_2=f\left(x_n+\frac{h}{2}, y_n+h \frac{q_1}{2}\right) q_3=f\left(x_n+\frac{h}{2}, y_n+h \frac{q_2}{2}\right) q_4=f\left(x_n+h, y_n+h q_3\right)$$
Euler 和 RK 方法的区别在于 Euler 方法利用了斜率或导数的信息 $y$ 在给定的时间步将解外推到下一个时间 步。另一方面，Runge-Kutta 方法在多个点利用”斜率”上的信息来推断末来时间步的解。

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