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A Convergence order of central finite difference scheme

  1. Nov 8, 2017 #1
    For example, when we solve simple 1D Poisson equation by finite difference method, why three point central difference scheme on uniform grid (attached image) is second order method for solution convergence?

    I understand why approximation of first derivative is second order (and that second derivative is also second order because cancelation of first order truncation error term on uniform grid), but I don’t understand why solution also converges with second order.

    Basically, this approximation is equivalent to fitting local quadratic (p=2) polynomial through three points which should theoretically yield third (p+1) order method?
     

    Attached Files:

  2. jcsd
  3. Nov 8, 2017 #2
    What is the differential equation that you are solving and what numerical scheme are you using?

    A finite difference is just an approximation of a derivative. By replacing the derivatives in a partial differential equation with their finite difference representations, you will get a large system of simultaneous equations. A classic example of this is the four point stencil for the two dimensional Laplace equation. This will produce a sparse diagonally banded matrix which can be solved very quickly using successive over-relaxation (SOR). There are other PDEs however, where finite difference approximations don't produce a system that is easy to solve. Non-linear PDE's are an obvious example since the system of equations cannot be cast as a matrix equation.
     
  4. Nov 9, 2017 #3
    Thx NFuller, but that wasn't what I had in mind. Here are some more details.
    I used example of linear 1D Poisson equation (smooth exact solution) and compared convergence of few numerical methods with different order of basis (interpolation) functions. I used B-spline functions as basis functions. Order of convergence was determined for each method and each basis function by plotting graph "dx - error norm" in log-log scale and slope of the curve determines convergence order of method.
    Theoretically convergence order depends of polynomial order of basis function (Taylor series expansion), and for Galerkin method (classical FEM) there is full convergence order, i.e. if "p" is order of basis function than solution converges with order "p+1", while first derivative converges with order "p" and second derivative with order "p-1". For collocation method that is not case and there is reduction in convergence order. Now I'm trying to figure way.
    So I used also Finite difference method (FDM) with 3 point stencil which is equivalent to approximation of second derivative by local polynomial of order p=2 (i.e. quadratic) and solution converges with order 2. By looking into Taylor expansion it is clear that first derivative approximation converges with order 2, and that second derivative converge also with order 2 (for uniform/symmetric grid and even-order polynomials as in this case). Now I'm looking for more detailed explanation why central difference scheme on three point stencil has same convergence order for solution as for first two derivatives.
     
  5. Nov 10, 2017 #4
    $$f(x+\Delta x)=f(x)+f'(x)\Delta x+f''(x)\frac{(\Delta x) ^2}{2}+f'''(x)\frac{(\Delta x)^3}{6}+f''''(x)\frac{(\Delta x)^4}{24}$$
    $$f(x-\Delta x)=f(x)-f'(x)\Delta x+f''(x)\frac{(\Delta x) ^2}{2}-f'''(x)\frac{(\Delta x)^3}{6}+f''''(x)\frac{(\Delta x)^4}{24}$$
    $$\frac{f(x+\Delta x)-2f(x)+f(x-\Delta x)}{(\Delta x)^2}=f''(x)+f''''(x)\frac{(\Delta x)^2}{12}$$
     
  6. Nov 10, 2017 #5
    Thx, Chestermiller, but I'm interested how can I relate this with convergence of discretization error for approximation of boundary value problem with FDM, i.e., convergence order of numerical solution, and not truncation error for approximation of derivative.
     
  7. Nov 10, 2017 #6
    In my judgment (and experience), it really boils down to the same thing.
     
  8. Nov 10, 2017 #7
    I have posted same question here:
    https://www.cfd-online.com/Forums/m...e-point-central-finite-difference-scheme.html

    If we use classical Galerkin FEM and quadratic Lagrangian polynomials we will get that solution converges by order 3. If we differentiate numerical solution and compare with exact first derivative of solution we will get that 1st derivative converges by order 2, and 2nd derivative will converge with order 1. It means that in Galerkin FEM convergence order depends directly on order of Lagrangian polynomial (I am assuming ideal case when solution is sufficiently regular).

    Now I'm trying to relate this with convergence of FDM. Apparently we are dealing with quadratic polynomial (or even of degree 4; because of symmetry for second derivative approximation) but our FDM is only 2nd order method. I can't figure why. I know that collocation method for same basis function will have reduced order when compared with Galerkin.

    Is there some order reduction in FDM because its nature of approximation?
     
  9. Nov 10, 2017 #8
    I don't know how to answer this except to say that I have solved many, many, many PDE's (parabolic, elliptic, Poisson) in my day using central differences for the spatial derivatives (FDM), and have always found that the solutions converged with 2nd order accuracy in the spatial increments. That is, the errors were of order ##(\Delta x)^2##.
     
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