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Motivation: Partial differential equations (PDEs) are broadly classified into parabolic, elliptic, and hyperbolic types.
Parabolic PDEs capture diffusion processes such as transient heat conduction.
Elliptic PDEs describe steady-state fields, exemplified by incompressible flow governed by stream function formulations.
Hyperbolic PDEs govern wave propagation, where information travels along characteristic lines.
Parabolic PDE: Transient Heat Conduction in a Wall
We consider a 1 ft thick wall initially at 100 °F, with both surfaces suddenly raised and held at 300 °F. The governing equation is the unsteady one-dimensional heat conduction equation:
This problem serves as a benchmark for parabolic PDE solvers, highlighting the trade-offs between explicit and implicit schemes in terms of stability, accuracy, and computational cost.
Elliptic PDE: Streamfunction in a Chamber
We consider a steady, incompressible, inviscid 2D flow in a chamber with specified inlet, outlet, and wall boundary conditions. Introducing the streamfunction ψ\psiψ, the governing equation reduces to the Laplace equation:
This problem serves as a benchmark for elliptic PDE solvers, illustrating boundary-value formulations, iterative solution strategies (Gauss-Seidel, SOR, ADI), and convergence behavior under different relaxation parameters and initial data distributions.
Hyperbolic PDE: 1D Wave Propagation in a Tube
We consider a triangular wave propagating in a closed-end tube with a constant wave speed of 200 m/s. The governing equation is the linear advection equation:
This problem serves as a benchmark for hyperbolic PDE solvers, highlighting the effects of numerical schemes (Upwind, Lax-Wendroff, BTCS) on wave propagation, dispersion, and stability.
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