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[WSS2020] Wolfram Models as Discretization Methods for Numerical PDE Solver

Yanal Marji, Minerva Schools at KGI

Posted 1 month ago

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In[]:= Wolfram Community Post Abstract: The Wolfram Models are a discrete spacetime formalism represented by hypergraphs and causal graphs generated by simple computational rules. Given that these models represent spacetime, they can readily be used as spatial and temporal discretizations for numerically computing Partial Differential Equations (PDEs), which can prove useful in cases where the discretizations are too computationally expensive or difficult to produce. This project is a preliminary study on ways the hypergraphs with interesting limiting geometries can be converted into a mesh for solving PDEs. Future Work: There are two primary directions this work can be expanded upon. The first being the incorporation of the causal graphs as a temporal discretization implemented through changing the topology of the spatial mesh with every iteration. The second is implementing a mesh-free method that can be used directly on the hypergraphs. When considering mathematical tools that are used in modelling, PDEs are arguably the most ubiquitous. Ironically enough, though, they are incredibly difficult, if not impossible, to solve analytically, which forces us to turn to numerical algorithms to compute them. However, these algorithms come with their own set of challenges, mainly because the PDEs have to be converted from a continuous to a discrete domain, this process is known as discretization. This is where the Wolfram models come in to save the day. Given that these models are already discrete, and limit to interesting geometries that would otherwise be difficult to generate, they can be used as a “ready-made” tool to handle this. In order to start this project properly, we begin by solving a simple PDE: The Poisson Equation. With our PDE chosen, we can now explore possible hypergraphs to solve our equation on. Consider the following hypergraph: g1=ResourceFunction["WolframModel"][{{x,y,y},{z,x,u}}{{y,v,y},{y,z,v},{u,v,v}},{{0,0,0},{0,0,0}},1000,"FinalState"]G1=ResourceFunction["WolframModelPlot"][g1] In[]:= This looks like a pretty interesting graph to start with. The way we'll be solving our equation on it is using the built-in NDSolveValue function in Mathematica, which means we need to convert this hypergraph into a mesh. The process is as follows: 1- Convert the hypergraph into an ordinary graph 2- Coordinatize the graph by embedding it into a Cartesian Plane and scale the output 3- Convert it into a mesh using built-in Mathematica functionality. Gr=ResourceFunction["HypergraphToGraph"][g1]C1=GraphEmbedding[Gr]C1=C1/(Mean[C1[[All,1]]])C1=#-{Mean[C1[[All,1]]],Mean[C1[[All,2]]]}&/@(C1)ListPlot[C1] In[]:= Out[]= Out[]= Mesh1=ResourceFunction["NonConvexHullMesh"][C1,0.2] In[]:= Out[]= And now we have a mesh ready to be used! The next step is to define and solve the PDE: ClearAll[x];ClearAll[y]op=-Laplacian[u[x,y],{x,y}]-5Bndry=DirichletCondition[u[x,y]==1,x≤0]ufun=NDSolveValue[{op==0,Bndry},u[x,y],{x,y}∈Mesh1]VertexWeightValues={}For[i=1,i<VertexCount[Gr]+1,i++,x=C1[[i]][[1]];y=C1[[i]][[2]];VertexWeightValues={VertexWeightValues,ufun}]VertexWeightValues=VertexWeightValues//FlattenWGraph1=Graph[Gr,VertexWeightVertexWeightValues]SetProperty[WGraph1,{VertexStyleThread[VertexList[WGraph1](ColorData["TemperatureMap"]/@Rescale[VertexWeightValues])]}] In[]:= Out[]= And this is the final result! We just solved the Poisson equation and stored the solution at each point in its corresponding vertex. The final graph is colored according to a temperature map, so the warmer the color, the higher the value of the PDE is at that point, pretty cool right? Now that we've established a procedure, we can extend it a bit further. Right now, we're just solving the PDE on a static hypergraph, meaning it's one single evolution. The next step would be generalizing the procedure into a function and then iteratively applying it over the hypergraph as it evolves. HypergraphPDESolver[g_,op_,bndry_]:=Module[{G,C,SC,Mesh,ufun,VVal,WGraph},G=ResourceFunction["HypergraphToGraph"][g];C=GraphEmbedding[G];SC=#-{Mean[C[[All,1]]],Mean[C[[All,2]]]}&/@(C);SC=SC/(Mean[C[[All,1]]]);Mesh=ResourceFunction["NonConvexHullMesh"][SC,0.2];ClearAll[x];ClearAll[y];ufun=NDSolveValue[{op==0,bndry},u[x,y],{x,y}∈Mesh];VVal={};For[i=1,i<VertexCount[G]+1,i++,x=SC[[i]][[1]];y=SC[[i]][[2]];VVal={VVal,ufun}];VVal=VVal//Flatten;WGraph=Graph[G,VertexWeightVVal];SetProperty[WGraph,{VertexSize4,VertexStyleThread[VertexList[WGraph](ColorData["TemperatureMap"]/@Rescale[VVal])]}]] In[]:= (SpatialEvolutionofGraphs)GraphList={}For[j=0,j<11,j++,g=ResourceFunction["WolframModel"][{{x,y,y},{z,x,u}}{{y,v,y},{y,z,v},{u,v,v}},{{1,2,2},{2,1,2}},500+(j100),"FinalState"];ClearAll[x];ClearAll[y];op=-Laplacian[u[x,y],{x,y}]-5;Bndry=DirichletCondition[u[x,y]==1,x≤0];GraphList={GraphList,HypergraphPDESolver[g,op,Bndry]}]GraphList=Flatten[GraphList] In[]:= {} Out[]=  , , , , , , , , , ,  With the generation of the solutions over multiple evolutions of the hypergraph, we finished a large portion of the work to be done, the only remaining part is incorporating time. The PDE we're going to be using for this next part is the 2D Burgers' equation, a quasilinear hyperbolic equation, and in the case with two spatial dimensions is represented by a system of coupled PDEs. This is significantly more difficult to solve than our first example, but still doable. We start by defining a function that generates a mesh, the same as the first part, as follows: ClearAll[x];ClearAll[y]GraphGenerator[g_]:=Module[{G,C,SC},G=ResourceFunction["HypergraphToGraph"][g];C=GraphEmbedding[G]; SC=SC/(Mean[C[[All,1]]]);SC=#-{Mean[C[[All,1]]],Mean[C[[All,2]]]}&/@(C); Return[G]] In[]:= K=GraphGenerator[ResourceFunction["WolframModel"][{{x,y,y},{z,x,u}}{{y,v,y},{y,z,v},{u,v,v}},{{0,0,0},{0,0,0}},1023,"FinalState"]] In[]:= VVals=First[Table[1,{i,32},{j,32}]//MatrixForm]UVals=First[Table[1,{i,32},{j,32}]//MatrixForm]VVals[[(3);;(11),(3);;11]]=Table[ConstantArray[2,9],{i,1,9}]UVals[[(3);;(11),(3);;11]]=Table[ConstantArray[2,9],{i,1,9}]GT1=Graph[K,VertexWeight(VVals//Flatten)]GraphList2={SetProperty[GT1,{VertexStyleThread[VertexList[GT1](ColorData["TemperatureMap"]/@Rescale[VVals//Flatten])]}]}Flatten[GraphList2] In[]:= dx=0.01dy=0.01dt=0.01For[i=1,i<6,i++,Un=UVals;Vn=VVals;UVals[[2;;31,2;;31]]=(Un[[2;;31,2;;31]]-(dt/dx)Un[[2;;31,2;;31]](Un[[2;;31,2;;31]]-Un[[2;;31,1;;30]])-(dt/dy)Vn[[2;;31,2;;31]](Un[[2;;31,2;;31]]-Un[[1;;30,2;;31]])+(0.01((dt/dx)^2))(Un[[2;;31,3;;32]]-(2Un[[2;;31,2;;31]]+Un[[2;;31,1;;30]]))+(0.01((dt/dy)^2))(Un[[3;;32,2;;31]]-(2Un[[2;;31,2;;31]])+Un[[1;;30,2;;31]]));VVals[[2;;31,2;;31]]=(Vn[[2;;31,2;;31]]-(dt/dx)Un[[2;;31,2;;31]](Vn[[2;;31,2;;31]]-Vn[[2;;31,1;;30]])-(dt/dy)Vn[[2;;31,2;;31]](Vn[[2;;31,2;;31]]-Vn[[1;;30,2;;31]])+(0.01((dt/dx)^2))(Vn[[2;;31,3;;32]]-(2Vn[[2;;31,2;;31]]+Vn[[2;;31,1;;30]]))+(0.01((dt/dy)^2))(Vn[[3;;32,2;;31]]-(2*Vn[[2;;31,2;;31]])+Vn[[1;;30,2;;31]]));WGraph=Graph[K,VertexWeight(VVals//Flatten)];GraphList2={GraphList2,SetProperty[WGraph,{VertexStyleThread[VertexList[WGraph](ColorData["TemperatureMap"]/@Rescale[VVals//Flatten])]}]}]Flatten[GraphList2] In[]:=  , , , , ,  Out[]=

POSTED BY: Yanal Marji

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