We first use the Euler method to get a y_n+1 in the Improved Euler method. Then we use another method to amend the y_n+1 to let it more similar to the real y_n+1.

We express this new Improved Euler method by these two formulas:

y_n+1 = y_n + hf (x_n , y_n )
y_n+1 = y_n + h*[f (x_n , y_n )+f (x_n+1 , y_n+1 )]/2

When use computer for calculation in some engineering field, you should show the relationship between ‘y’ and ‘x’. You need also give a ‘h’.

A example to use Improved Euler method for solving heat transfer problem:(Use Matlab)

Euler method used in heat transfer calculation

Question: As the figure shows. T_∞=1500℃，T_w=500 ℃, T₀=20℃, a thermocouple which diameter is 1mm exposed in the air. The surface coefficient of heat transfer h=200W(m²/K)，ε=0.2. Thermocouple oxidation will create energy at a rate of q=1000W/m³.  Determine the relation ship between temperature of thermocouple and time. [additional properties of thermocouple:ρ=8000kg/m³，c=0.38kJ/(kg*K)，λ=50W/(m*K)]

Answer: We use Improved Euler Question to solve this problem.  www.zhangpeng.info

The source code in Matlab here:

function [x,y]=eulerg(x,y,h,N)
x=zeros(1,N+1);
y=zeros(1,N+1);
x(1)=0;
y(1)=293;
for n=1:N
x(n+1)=x(n)+h;
ybar=y(n)+h*dyfun(x(n),y(n));
y(n+1)=y(n)+(h/2)*(dyfun(x(n),y(n))+dyfun(x(n),ybar));
end
plot(x,y,'red');
function m=dyfun(x,y)
m=1/304+75/19*(1773-y)+(6.804e-8)/304*(1773^4-y^4);

To create a new file in Matlab writing these codes and save it. Input some properties, for example eulerg(0,293,0.001,1000), and you will gain the relationship line between ‘y’ and ‘x’.

You can transfer this example in any other similar work process by just change some part of this example function. I’m glad to discuss with you if you have any questions.

在这种改进的尤拉格式中，先用尤拉方法求得一个初步的近似值，记y_n+1，这一步即预报，然后再通过梯形方法对这一预报值进行修正，得到较为精确的y_n+1值，这是一种热工数值计算中常用的方法。

这一方法可以表示为以下的两式：

• 预报: y_n+1 = y_n + hf (x_n , y_n )
• 校正: y_n+1 = y_n + h*[f (x_n , y_n )+f (x_n+1 , y_n+1 )]/2

在计算机求解的时候，我们需要给出一个y关于x的函数，同时设定步长h，这个函数不管是显式或者隐式的函数都可以。

Matlab中利用改进尤拉方法计算传热问题举例：

尤拉格式计算传热问题

题：薄壁物体温度响应的数值求解。如图所示，T_∞=1500℃，T_w=500 ℃。从某时刻起，T₀=20℃，直径1mm的热电偶结点置于高温气体中，气流对热电偶的表面换热系数 h=200W(m²/K)，结点表面发射率 ε=0.2，高温气流对金属的氧化作用使热电偶结点产生一定的热量，其强度为 q=1000W/m3，求热电偶结点的温度响应。结点材料：ρ=8000kg/m³，c=0.38kJ/(kg*K)，λ=50W/(m*K)

解：我们在这里使用改进尤拉方法进行热工数值计算。www.zhangpeng.info

Matlab源代码如下：

function [x,y]=eulerg(x,y,h,N)
x=zeros(1,N+1);
y=zeros(1,N+1);
x(1)=0;
y(1)=293;
for n=1:N
x(n+1)=x(n)+h;
ybar=y(n)+h*dyfun(x(n),y(n));
y(n+1)=y(n)+(h/2)*(dyfun(x(n),y(n))+dyfun(x(n),ybar));
end
plot(x,y,'red');
function m=dyfun(x,y)
m=1/304+75/19*(1773-y)+(6.804e-8)/304*(1773^4-y^4);

在 matlab 中新建文件写入代码保存后，输入例如：eulerg(0,293,0.001,1000) 即可以得出热电偶温度y随时间x的变化规律曲线。