2. Consider the initial value problem ut + auxxx = 0, x = [0, 1], u(x,0)...

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2. Consider the initial value problem ut + auxxx = 0, x = [0, 1], u(x,0) = Ok(x) = e²nikx a> 0 € R, and periodic boundary conditions. Suppose we use the following scheme to solve this PDE: At v+¹ = v +1 - a ² (Ax)3 (v₁ +2 - 3√5 +1 + 3v² − v ²7 -₁) [a] What is the exact solution to this problem? Describe its behavior. [b] Show that this scheme is consistent. [c] What modified equation does this scheme solve? [d] Based on the modified equation, describe the long term behavior of the numerical solution. Be as specific as possible. [e] Can you determine from your analysis why a cannot be less than zero for this scheme to be stable? Explain. Hint: Consider the solution of the modified equation. Can you suggest a way to modify the scheme so that it would be stable for a < 0? [f] Determine the stability region for a for this scheme. You do not have to plot it or reduce it algebraically. However, you should be able to determine a stability condition for At. Hint: Notice that the scheme is centered around the point x;+ [g] Does the semi-discrete version of this scheme constitute a stiff problem? Explain. 2. Consider the initial value problem ut + auxxx = 0, x = [0, 1], u(x,0) = Ok(x) = e²nikx a> 0 € R, and periodic boundary conditions. Suppose we use the following scheme to solve this PDE: At v+¹ = v +1 - a ² (Ax)3 (v₁ +2 - 3√5 +1 + 3v² − v ²7 -₁) [a] What is the exact solution to this problem? Describe its behavior. [b] Show that this scheme is consistent. [c] What modified equation does this scheme solve? [d] Based on the modified equation, describe the long term behavior of the numerical solution. Be as specific as possible. [e] Can you determine from your analysis why a cannot be less than zero for this scheme to be stable? Explain. Hint: Consider the solution of the modified equation. Can you suggest a way to modify the scheme so that it would be stable for a < 0? [f] Determine the stability region for a for this scheme. You do not have to plot it or reduce it algebraically. However, you should be able to determine a stability condition for At. Hint: Notice that the scheme is centered around the point x;+ [g] Does the semi-discrete version of this scheme constitute a stiff problem? Explain.

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a The exact solution to this problem is given by the solution of the differential equation u au xxx 0 This is a linear differential equation with a variable coefficient and so it has the general solut... View the full answer