differential equations - Has this implementation of FDM touched the speed limit of Mathematica?

Still, I'll use the implementation of the 1D FDTD method (you can simply understand it as a kind of explicit finite difference scheme for the Maxwell's equation) as the example. Just for completeness, here is the 1D Maxwell's equation:

$$\mu \frac{\partial H_y}{\partial t}=\frac{\partial E_z}{\partial x}$$ $$\epsilon \frac{\partial E_z}{\partial t}=\frac{\partial H_y}{\partial x}$$

and the corresponding finite difference equation:

$$H_y^{q+\frac{1}{2}}\left[m+\frac{1}{2}\right]\text{==}H_y^{q-\frac{1}{2}}\left[m+\frac{1}{2}\right]+\frac{\Delta _t}{\mu \Delta _x}\left(E_z^q[m+1]-E_z^q[m]\right)$$ $$E_z^{q+1}[m]==E_z^q[m]+\frac{\Delta _t}{\epsilon \Delta _x}\left(H_y^{q+\frac{1}{2}}\left[m+\frac{1}{2}\right]-H_y^{q+\frac{1}{2}}\left[m-\frac{1}{2}\right]\right)$$

The toy code I've repeatedly used in several posts implementing the difference scheme is:

ie = 200;
ez = ConstantArray[0., {ie + 1}];
hy = ConstantArray[0., {ie}];


fdtd1d = Compile[{{steps}}, 
   Module[{ie = ie, ez = ez, hy = hy}, 
    Do[ez[[2 ;; ie]] += (hy[[2 ;; ie]] - hy[[1 ;; ie - 1]]);
     ez[[1]] = Sin[n/10];
     hy[[1 ;; ie]] += (ez[[2 ;; ie + 1]] - ez[[1 ;; ie]]), {n, 
      steps}]; ez]];

result = fdtd1d[10000]; // AbsoluteTiming

Notice that constants like $\mu$, $\Delta _t$ are omitted for simplicity.

Personally I think it's a typical example for the implementation of finite difference method (FDM), so here's the question: has this piece of code (at least almost) touched the speed limit of Mathematica? In fact several months ago, I found that if the code is rewrited with Julia, it'll be 4 times faster.

Indeed, I know this might be the case that one should use the best-suited tool for a specific job, but since I've already gained some stupid pride on using Mathematica and am unwilling to spend time to learn a new programming language (Wolfram is almost my first programming language, I used to learn some VB, but already gave it back to my teacher when started to use Mathematica), I still want to make sure if the Mathematica version of the code can be faster.

If it's the limitation, I'd like to know why there's such a big difference.

Any help would be appreciated.

Answer

Okay, this is a bit of an embarassment.

Here is a very small modification of the original code. I simply made explicit option settings, made a denominator to Sin explicitly real, that kind of thing. My tests show the same timing as the original, give or take an iota.

ie = 200;
ez = ConstantArray[0., {ie + 1}];

hy = ConstantArray[0., {ie}];

fdtd1d = Compile[{{steps}}, 
   Module[{ie = ie, ez = ez, hy = hy}, 
    Do[ez[[2 ;; ie]] += (hy[[2 ;; ie]] - hy[[1 ;; ie - 1]]);
     ez[[1]] = Sin[n/10.];
     hy[[1 ;; ie]] += (ez[[2 ;; ie + 1]] - ez[[1 ;; ie]]), {n, 
      steps}]; ez], 
   CompilationOptions -> {"InlineExternalDefinitions" -> True}, 
   "CompilationTarget" -> "C", "RuntimeOptions" -> "Speed"];

I'll beef up the example a factor of 10.

result = fdtd1d[100000]; // AbsoluteTiming

(* Out[172]= {0.555320, Null} *)

Now we remove the spans and replace with inner loops. No vectorization, that is.

fdtd1d2 = Compile[{{steps}}, Module[{ie = ie, ez = ez, hy = hy},
    Do[
     Do[ez[[j]] += (hy[[j]] - hy[[j - 1]]), {j, 2, ie}];

     Do[ez[[1]] = Sin[n/10.];
      hy[[j - 1]] += (ez[[j]] - ez[[j - 1]]), {j, 2, ie + 1}], {n, 
      steps}]; ez], 
   CompilationOptions -> {"InlineExternalDefinitions" -> True}, 
   "CompilationTarget" -> "C", "RuntimeOptions" -> "Speed"];

result2 = fdtd1d2[100000]; // AbsoluteTiming
result2 == result

(* Out[174]= {0.179435, Null}


Out[175]= True *)

So that's a factor of 3. I guess I need to report this as a suggestion to revisit vector operations in Compile.

--- edit ---

It is of course slightly faster to take the assignment of ez[[1]] =... out of the second inner loop (sound of hand smacking forehead). Also it turns out to be slightly faster to reduce the index arithmetic in the second loop. I also took the assignment of the constant, ie, out of the Module variables and made it really constant using With; this does not seem to affect timing one way or the other.

fdtd1d3 = With[{ie = ie}, Compile[{{steps, _Integer}},
    Module[
     {ez = ez, hy = hy},
     Do[

      ez[[1]] = Sin[n/10.];
      Do[ez[[j]] += (hy[[j]] - hy[[j - 1]]), {j, 2, ie}];
      Do[
       hy[[j]] += (ez[[j + 1]] - ez[[j]]), {j, 1, ie}],
      {n, steps}]; ez],
    CompilationOptions -> {"InlineExternalDefinitions" -> True}, 
    "CompilationTarget" -> "C", "RuntimeOptions" -> "Speed"]];

result3 = fdtd1d3[100000]; // AbsoluteTiming
result3 === result2


(* Out[107]= {0.135636, Null}

Out[108]= True *)

So that's a modest improvement.

--- end edit ---

--- edit #2 ---

Per suggestion by @s0rce, we'll use Compile`GetElement instead of Part.

fdtd1d4 = Compile[{{steps, _Integer}},

   Module[
    {ie = ie, ez = ez, hy = hy},
    Do[
     ez[[1]] = Sin[n/10.];
     Do[ez[[j]] += (Compile`GetElement[hy, j] - 
         Compile`GetElement[hy, j - 1]), {j, 2, ie}];
     Do[
      hy[[j]] += (Compile`GetElement[ez, j + 1] - 
         Compile`GetElement[ez, j]), {j, 1, ie}],
     {n, steps}]; ez],

   CompilationOptions -> {"InlineExternalDefinitions" -> True}, 
   "CompilationTarget" -> "C", "RuntimeOptions" -> "Speed"];

result4 = fdtd1d4[100000]; // AbsoluteTiming
result4 === result2

(* Out[122]= {0.076532, Null}

Out[123]= True *)

Now that's progress.

--- end edit #2 ---