Implicit Methods

PDEs · Axiom Academy

Understanding BTCS, unconditional stability, and solving tridiagonal systems 1. BTCS (Backward Time Central Space) Scheme The BTCS scheme uses a backward difference in time and central difference in space . For the heat equation, we approximate derivatives at time level n+1 instead of n : This means we evaluate the spatial derivative at the unknown future time step. Rearranging gives us an equation that couples all spatial points at the new time level together. FTCS (Explicit): Uses known values at time n to directly compute values at time n+1 BTCS (Implicit): Creates equations involving unknown values at time n+1 , requiring solving a system 2. System of Equations at Each Time Step When we apply BTCS to all interior points in our spatial grid, we get a system of linear equations . For the heat equation with coefficient r = αΔt/Δx², the discretized form becomes: With boundary conditions u₀ and uₙ known, this creates a system of equations for the interior points u₁, u₂, ..., uₙ₋₁. The system has a special structure called tridiagonal because each equation only involves three adjacent points. The most powerful feature of BTCS is unconditional stability . Unlike FTCS which requires r ≤ 0.5, BTCS is stable for any choice of Δt and Δx: This means we can use much larger time steps without numerical instability. The solution remains bounded and well-behaved regardless of the time step size, though accuracy still depends on discretization error.

This is the written version of the interactive lesson above. See the full PDEs course.