Solution Interpolation Method for Highly Oscillating Hyperbolic Equations

Figure 6

Time evolution of the errors in numerical solutions for 𝑢_𝑡𝑡=(2+sin(−𝑥/𝜀))⁻¹𝑢_𝑥𝑥,0≤𝑥≤1. The exact solution 𝑢=(1+𝑥)𝑡²+𝑥𝑡+𝑥+2(2𝜀³+𝜀𝑥+𝑥²+𝑥³/3−2𝜀³cos(𝑥/𝜀)−𝜀²(1+𝑥)sin(𝑥/𝜀)) is used to generate the initial/boundary conditions and measure the errors.

(a) 𝜀=1

(b) 𝜀=0.01